Right Buchberger algorithm over bijective skew PBW extensions
aa r X i v : . [ m a t h . R A ] J a n Right Buchberger algorithm over bijective skew
P BW extensions
William FajardoSeminario de Álgebra Constructiva - SAC Departamento de MatemáticasUniversidad Nacional de Colombia, Bogotá, [email protected]
Abstract
In this paper we present a right version of the algorithms developed for to compute Gröbnerbases over bijective skew
PBW extensions in the left case given in [ ] . In particular, we adaptthe theory of reduction and we build a right division algorithm and generate a right version ofBuchberger algorithm over bijective skew PBW extensions, finally we illustrate some examplesusing the SPBWE.lib library implemented in Maple (see [ ] , [ ] ). It is important to note that thedevelopment of this theory is fundamental to complete the SPBWE.lib library and to be able todevelop many of the homological applications that arise as result of obtaining the right Gröbnerbases over skew
PBW extensions.
Key words and phrases.
Noncommutative computational algebra, skew
PBW extensions, Buch-berger algorithm, Gröbner bases,
SPBWE.lib library, Maple.2021
Mathematics Subject Classification.
Primary: 16Z05. Secondary: 16D40, 15A21.
P BW extensions
In this section we introduce the bijective skew P BW extensions whose are the fundamental topic inthis paper. Skew
P BW extensions include well known classes of Ore algebras, operator algebras andalso a lot of quantum rings and algebras. The skew
P BW extensions have been extensively studied,in particular for to complete compilation of these studies, see [ ] . The skew P BW extensions arebeing implemented in the
SPBWE.lib library developed in Maple, the main purpose of the paper isto illustrate the construction of the theory needed to develop the right Gröbner theory and generatetheir respective algorithms implemented in Maple through
SPBWE.lib library., i.e., implementingthe division algorithm and Buchberger algorithm in the right case, similarly since in Fajardo [ ] and [ ] , Fajardo-Lezama [ ] , Simson [ ] and Simson-Wojewodzki [ ] . Definition 1.1.
Let R and A be rings, we say that A is a skew P BW extension of R ( also called σ -P BWextension ) , if the following conditions hold: (i) R ⊆ A. There exist finitely many elements x , . . . , x n ∈ A such that A is a left R-free module with basis
Mon ( A ) : = Mon { x , . . . , x n } = { x α : = x α · · · x α n n | α = ( α , . . . , α n ) ∈ N n } . (iii) For every ≤ i ≤ n and r ∈ R − { } there exists c i , r ∈ R − { } such thatx i r − c i , r x i ∈ R . (1.1)(iv) For every ≤ i , j ≤ n there exists c i , j ∈ R − { } such thatx j x i − c i , j x i x j ∈ R + Rx + · · · + Rx n . (1.2) Under these conditions we will write A = σ ( R ) 〈 x , . . . , x n 〉 , and R will be called the ring of coefficientsof the extension. Remark 1.2.
Each element f ∈ A − { } has a unique representation in the form f = c X + · · · + c t X t ,with c i ∈ R − { } and X i ∈ Mon ( A ) , 1 ≤ i ≤ t .The following proposition (see [ ] , Proposition 1.1.3) justifies the notation given in definition ofthe skew P BW extensions.
Proposition 1.3.
Let A = σ ( R ) 〈 x , . . . , x n 〉 be a skew P BW extension of R. Then, for ≤ i ≤ n, thereexist an injective ring endomorphism σ i : R → R and a σ i -derivation δ i : R → R such thatx i r = σ i ( r ) x i + δ i ( r ) ,for every r ∈ R. Definition 1.4.
Let A = σ ( R ) 〈 x , . . . , x n 〉 be a skew P BW extension of R. A is called bijective if σ i isbijective for every ≤ i ≤ n and c i , j is invertible for any ≤ i , j ≤ n. Definition 1.5.
Let A = σ ( R ) 〈 x , . . . , x n 〉 be a skew P BW extension of R , with endomorphisms σ i , 1 ≤ i ≤ n. We will use the following notation. (i) For α = ( α , . . . , α n ) ∈ N n , σ α : = σ α · · · σ α n n , | α | : = α + · · · + α n and if A is bijective σ − α : = σ − α n n · · · σ − α . Moreover, if β = ( β , . . . , β n ) ∈ N n , then α + β : = ( α + β , . . . , α n + β n ) . (ii) For X = x α ∈ Mon ( A ) , exp ( X ) : = α and deg ( X ) : = | α | . (iii) Let = f ∈ A; if f = c X + · · · + c t X t , with X i ∈ Mon ( A ) and c i ∈ R − { } , then deg ( f ) : = max { deg ( X i ) } ti = .The following characterization of skew P BW extensions was given in [ ] . Theorem 1.6.
Let A be a ring of a left polynomial type over R w.r.t. { x , . . . , x n } . A is a skew P BWextension of R if and only if the following conditions hold: (a) For every x α ∈ Mon ( A ) and every = r ∈ R there exist unique elements r α : = σ α ( r ) ∈ R − { } and p α , r ∈ A such that x α r = r α x α + p α , r , (1.3) where p α , r = or deg ( p α , r ) < | α | if p α , r = . Moreover, if r is left invertible, then r α is leftinvertible. For every x α , x β ∈ Mon ( A ) there exist unique elements c α , β ∈ R and p α , β ∈ A such thatx α x β = c α , β x α + β + p α , β , (1.4) where c α , β is left invertible, p α , β = or deg ( p α , β ) < | α + β | if p α , β = . Remark 1.7. (i) If a left inverse of c α , β is c ′ α , β . We observe that if in (b) of the previous theorem α = β = c α , β = c ′ α , β = θ , γ , β ∈ N n and c ∈ R , then we have the following identities: σ θ ( c γ , β ) c θ , γ + β = c θ , γ c θ + γ , β , (1.5) σ θ ( σ γ ( c )) c θ , γ = c θ , γ σ θ + γ ( c ) . (1.6)(iii) From Theorem 1.6 we get also that if A is bijective, then c α , β is invertible for any α , β ∈ N n . Mon ( A m ) In this section we will compile some results taken from [ ] that will be used in the theory of reductionand the theory of Grobner for the right case. Definition 2.1. In Mon ( A ) we definex α (cid:23) x β ⇐⇒ x α = x β orx α = x β but | α | > | β | orx α = x β , | α | = | β | but ∃ i with α = β , . . . , α i − = β i − , α i > β i . It is clear that this is a total order on
Mon ( A ) called deglex order. If x α (cid:23) x β but x α = x β , we writex α ≻ x β . Each element f ∈ A − { } can be represented in a unique way as f = c x α + · · · + c t x α t ,with c i ∈ R − { } , ≤ i ≤ t, and x α ≻ · · · ≻ x α t . We say that x α is the leader monomial of f andwe write lm ( f ) : = x α ; c is the leader coefficient of f , lc ( f ) : = c , and c x α is the leader term of fdenoted by lt ( f ) : = c x α . If f = , we define lm ( ) : =
0, lc ( ) : =
0, lt ( ) : = , and we set X ≻ forany X ∈ Mon ( A ) . P BW extensions
Let A = σ ( R ) 〈 x , . . . , x n 〉 be a skew P BW extension of R and let (cid:23) be a total order defined onMon ( A ) . If x α (cid:23) x β but x α = x β we will write x α ≻ x β . x β (cid:22) x α means that x α (cid:23) x β . Let f = A , if f = c X + · · · + c t X t ,3ith c i ∈ R −{ } and X ≻ · · · ≻ X t are the monomials of f , then lm ( f ) : = X is the leading monomial of f , lc ( f ) : = c is the leading coefficient of f and lt ( f ) : = c X is the leading term of f . If f =
0, wedefine lm ( ) : =
0, lc ( ) : =
0, lt ( ) : =
0, and we set X ≻ X ∈ Mon ( A ) . Thus, we extend (cid:23) to Mon ( A ) ∪ { } . Definition 2.2.
Let (cid:23) be a total order on
Mon ( A ) , it is said that (cid:23) is a monomial order on Mon ( A ) ifthe following conditions hold: (i) For every x β , x α , x γ , x λ ∈ Mon ( A ) x β (cid:23) x α ⇒ lm ( x γ x β x λ ) (cid:23) lm ( x γ x α x λ ) . (ii) x α (cid:23) , for every x α ∈ Mon ( A ) . (iii) (cid:23) is degree compatible, i.e., | β | ≥ | α | ⇒ x β (cid:23) x α . Monomial orders are also called admissible orders . It is worth noting that every monomial orderon Mon ( A ) is a well order. Thus, there are not infinite decreasing chains in Mon ( A ) . From now onwe will assume that Mon ( A ) is endowed with some monomial order. Definition 2.3.
Let x α , x β ∈ Mon ( A ) , we say that x α divides x β , denoted by x α | x β , if there existsx γ , x λ ∈ Mon ( A ) such that x β = lm ( x γ x α x λ ) . We will say also that any monomial x α ∈ Mon ( A ) divides the polynomial zero. The condition (iii) of Definition 2.2 is needed in the proof of the following proposition (see [ ] ,Proposition 13.1.4), and this one will be used in right Division Algorithm (Theorem 3.8). Proposition 2.4.
Let A a bijective skew PBW extension and x α , x β ∈ Mon ( A ) and f , g ∈ A − { } . Then, (a) lm ( x α g ) = lm ( x α lm ( g )) = x α + exp ( lm ( g )) , i.e., exp ( lm ( x α g )) = α + exp ( lm ( g ) . In particular, lm ( lm ( f ) lm ( g )) = x exp ( lm ( f ))+ exp ( lm ( g )) , i.e., exp ( lm ( lm ( f ) lm ( g ))) = exp ( lm ( f )) + exp ( lm ( g )) and lm ( x α x β ) = x α + β , i . e ., exp ( lm ( x α x β )) = α + β . (2.1)(b) The following conditions are equivalent: (i) x α | x β . (ii) There exists a unique x θ ∈ Mon ( A ) such that x β = lm ( x θ x α ) = x θ + α and hence β = θ + α . (iii) There exists a unique x θ ∈ Mon ( A ) such that x β = lm ( x α x θ ) = x α + θ and hence β = α + θ . (iv) β i ≥ α i for ≤ i ≤ n, with β : = ( β , . . . , β n ) and α : = ( α , . . . , α n ) . Remark 2.5. (cid:23) a monomial order on Mon ( A ) ; if there exists f = x γ c + · · · + x γ t c t ∈ A − { } such that x β = x α f , then by Proposition 2.4, x β = x α + γ , i.e., x α | x β . On the other hand, if x β = f x α ,then x β = t X i = x γ i c i x α = t X i = x γ i ( x α r α + p α , c i ) = t X i = x γ i + α c γ i , α r α + t X i = p γ i , α r α + x γ i p α , c i .Then x β = x α + γ , i.e., x α | x β .(ii) We note that there exists a least common multiple of two elements of Mon ( A ) : in fact, let x α , x β ∈ Mon ( A ) , then lcm ( x α , x β ) = x γ ∈ Mon ( A ) , where γ = ( γ , . . . , γ n ) with γ i : = max { α i , β i } for each 1 ≤ i ≤ n . Mon ( A m ) We will often represent the elements of A m also as row vectors, if this does not cause confusion. Werecall that the canonical basis of A m is e = (
1, 0, . . . , 0 ) , e = (
0, 1, 0, . . . , 0 ) , . . . , e m = (
0, 0, . . . , 1 ) . Definition 2.6.
A monomial in A m is a vector X = X e i , where X = x α ∈ Mon ( A ) and ≤ i ≤ m, i.e., X = X e i = (
0, . . . , X , . . . , 0 ) ,where X is in the i-th position, named the index of X , ind ( X ) : = i. A term is a vector c X , where c ∈ R.The set of monomials of A m will be denoted by Mon ( A m ) . Let Y = Y e j ∈ Mon ( A m ) , we say that X divides Y if i = j and X divides Y . We will say that any monomial X ∈ Mon ( A m ) divides the null vector . Theleast common multiple of X and Y , denoted by lcm ( X , Y ) , is if i = j, and U e i , where U = lcm ( X , Y ) ,if i = j. Finally, we define exp ( X ) : = exp ( X ) = α and deg ( X ) : = deg ( X ) = | α | . We now define monomial orders on Mon ( A m ) . Definition 2.7.
A monomial order on
Mon ( A m ) is a total order (cid:23) satisfying the following three condi-tions: (i) lm ( x β x α ) e i (cid:23) x α e i , for every monomial X = x α e i ∈ Mon ( A m ) and any monomial x β in Mon ( A ) . (ii) If Y = x β e j (cid:23) X = x α e i , then lm ( x γ x β ) e j (cid:23) lm ( x γ x α ) e i for every monomial x γ ∈ Mon ( A ) . (iii) (cid:23) is degree compatible, i.e., deg ( X ) ≥ deg ( Y ) ⇒ X (cid:23) Y .If X (cid:23) Y but X = Y we will write X ≻ Y . Y (cid:22) X means that X (cid:23) Y . From the above definition we have that every monomial order on Mon ( A m ) is a well order. Nextwe give a monomial order (cid:23) on Mon ( A ) , we can define two natural orders on Mon ( A m ) . Definition 2.8.
Let X = X e i and Y = Y e j ∈ Mon ( A m ) . The TOP ( term over position ) order is defined by X (cid:23) Y ⇐⇒ X (cid:23) YorX = Y and i > j .(ii) The TOPREV order is defined by X (cid:23) Y ⇐⇒ X (cid:23) YorX = Y and i < j . Remark 2.9. (i) Note that with TOP we have e m ≻ e m − ≻ · · · ≻ e and e ≻ e ≻ · · · ≻ e m for TOPREV.(ii) The POT (position over term) and POTREV orders defined in [ ] and [ ] for modules overclassical polynomial commutative rings are not degree compatible.We fix a monomial order on Mon ( A ) , let f = be a vector of A m , then we may write f as a sumof terms in the following way f = c X + · · · + c t X t ,where c , . . . , c t ∈ R − X ≻ X ≻ · · · ≻ X t are monomials of Mon ( A m ) . Definition 2.10.
Let f : = c X + · · · + c t X t ∈ A m where c , . . . , c t ∈ R − X ≻ X ≻ · · · ≻ X t monomialsof Mon ( A m ) and X i : = x γ i e j i with γ i ∈ N n . Let g ∈ A, then we define f g : = c x γ g e j + · · · + c t x γ t g e j t . Which is in A m . Remark 2.11.
With the notation of above definition, we have that exp ( lm ( f x α )) = exp ( lm ( f )) + α .In fact, as ≻ is monomial order on Mon ( A m ) , then lm ( x γ x α ) e j ≻ lm ( x γ k x α ) e j k for each 2 ≤ k ≤ t ,thus, lm ( f x α ) = lm ( x γ x α ) e j so, exp ( lm ( f x α )) = γ + α = exp ( lm ( f ))+ α . Hence, lc ( f x α ) = c c γ , α = lc ( f ) c γ , α . Definition 2.12.
With the above notation, we say that (i) lt ( f ) : = c X is the leading term of f . (ii) lc ( f ) : = c is the leading coefficient of f . (iii) lm ( f ) : = X is the leading monomial of f . For f = we define lm ( ) = , lc ( ) =
0, lt ( ) = , and if (cid:23) is a monomial order on Mon ( A m ) ,then we define X ≻ for any X ∈ Mon ( A m ) . So, we extend (cid:23) to Mon ( A m ) S { } .6 Right reduction in A m In this section we present the fundamental topics of reduction theory for right submodules of A m when A is a bijective skew P BW extension. This theory was studied in the bijective general case forleft modules (see [ ] ), here we adapt these ideas.We will assume some natural computational conditions on R . Definition 3.1.
A ring R is right Gröbner soluble ( RGS ) if the following conditions hold: (i) R is right Noetherian. (ii)
Given a , r , . . . , r m ∈ R there exists an algorithm which decides whether a is in the right idealr R + · · · + r m R, and if so, find b , . . . , b m ∈ R such that a = r b + · · · + r m b m . (iii) Given r , . . . , r m ∈ R there exists an algorithm which finds a finite set of generators of the rightR-module
Syz rR [ r · · · r m ] : = { ( b , . . . , b m ) ∈ R m | r b + · · · + r m b m = } . Remark 3.2.
The three above conditions imposed to R are needed in order to guarantee a rightGröbner theory in the rings of coefficients, in particular, to have an effective solution of the member-ship problem in R (see (ii) in Definition 3.3 below). From now on in this paper we will assume that A = σ ( R ) 〈 x , . . . , x n 〉 is a skew P BW extension of R , where R is a RGS ring and Mon ( A ) is endowedwith some monomial order.The reduction process in A m is defined as follows. Definition 3.3.
Let F be a finite set of non-zero vectors of A m , and let f , h ∈ A m , we say that f reducesto h by F in one step, denoted f F −→ h , if there exist elements f , . . . , f t ∈ F and r , . . . , r t ∈ R such that (i) lm ( f i ) | lm ( f ) , ≤ i ≤ t, i.e., ind ( lm ( f i )) = ind ( lm ( f )) and there exists x α i ∈ Mon ( A ) such that β i + α i = exp ( lm ( f )) with β i : = exp ( lm ( f i )) . (ii) lc ( f ) = lc ( f ) σ β ( r ) c f , α + · · · + lc ( f t ) σ α t ( r t ) c f t , α t , where c f i , α i : = c β i , α i . (iii) h = f − P ti = f i r i x α i .We say that f reduces to h by F , denoted f F −→ + h , if and only if there exist vectors h , . . . , h t − ∈ A m such that f F −−−−→ h F −−−−→ h F −−−−→ · · · F −−−−→ h t − F −−−−→ h . f is reduced ( also called minimal ) w.r.t. F if f = or there is no one step reduction of f by F , i.e., oneof the first two conditions of Definition 3.3 fails. Otherwise, we will say that f is reducible w.r.t. F . If f F −→ + h and h is reduced w.r.t. F , then we say that h is a remainder for f w.r.t. F . Remark 3.4.
Related to the previous definition we have the following remarks:(i) By Theorem 1.6, the coefficients c f i , α i in the previous definition are unique and satisfy x exp ( lm ( f i )) x α i = c f i , α i x exp ( lm ( f i ))+ α i + p f i , α i ,where p f i , α i = ( lm ( p f i , α i )) < | exp ( lm ( f i )) + α i | , 1 ≤ i ≤ t .7ii) lm ( f ) ≻ lm ( h ) and f − h ∈ 〈 F 〉 , where 〈 F 〉 is the right submodule of A m generated by F .(iii) The remainder of f is not unique.(vi) By definition we will assume that F −→ .(v) lt ( f ) = t X i = lt ( lt ( f i ) r i x α i ) ,From the reduction relation we get the following interesting properties. Proposition 3.5.
Let A be a bijective skew P BW extension. Let f , h ∈ A m , θ ∈ N n and F = { f , . . . , f t } be a finite set of non-zero vectors of A m . Then, (i) If f F −→ h , then there exists p ∈ A m with p = or lm ( f x θ ) ≻ lm (p ) such that f x θ + p F −→ h x θ . (ii) If f F −→ + h and p ∈ A is such that p = or lm ( h ) ≻ lm ( p ) , then f + p F −→ + h + p . (iii) If f F −→ + h , then there exists p ∈ A m with p = or lm ( f x θ ) ≻ lm ( p ) such that f x θ + p F −→ + x θ h . (iv) If f F −→ + , then there exists p ∈ A m with p = or lm ( f x θ ) ≻ lm ( p ) such that f x θ + p F −→ + .Proof. (i) If f = , then h = = p . Let f = and lm ( f ) : = x λ ; then there exist f , . . . , f t ∈ F and r . . . , r t ∈ R such that lm ( f i ) | lm ( f ) , for 1 ≤ i ≤ t , i.e., ind ( lm ( f i )) = ind ( lm ( f )) and thereexists x α i ∈ Mon ( A ) such that λ = α i + exp ( lm ( f i )) . Moreover,lc ( f ) = lc ( f ) σ β ( r ) c β , α + · · · + lc ( f t ) σ β t ( r t ) c β t , α t with β i : = exp ( lm ( f i )) and h = f − P ti = f i r i x α i . We note that ind ( lm ( f )) = ind ( lm ( f x θ )) andexp ( f x θ ) = θ + λ , so lm ( f i ) | lm ( f x θ ) , with θ + λ = ( θ + α i ) + β i ;we observe that lc ( f x θ ) = lc ( f ) c λ , θ = t X i = lc ( f i ) σ β i ( r i ) c β i , α i c λ , θ ,and by Remark 1.7 lc ( f x θ ) = t X i = lc ( f i ) σ β i ( r i ) c β i , α i c α i + β i , θ = t X i = lc ( f i ) σ β i ( r i ) σ β i ( c α i , θ ) c β i , α i + θ = t X i = lc ( f i ) σ β i ( r i c α i , θ ) c β i , α i + θ t X i = lc ( f i ) σ β i ( r ′ i ) c β i , α i + θ ,where r ′ i : = r i c α i , θ . Moreover, h x θ = f x θ − t X i = f i r i x α i x θ = f x θ − t X i = f i r i c α i , θ x α i + θ + p = f x θ + p − t X i = f i r ′ i x α i + θ where p : = P ti = ( − f i ) r i p α i , θ ; note that p = or deg ( p ) < | θ + α i + β i | = | θ + λ | = d e g ( f x θ ) ,so lm ( f x θ ) ≻ lm ( p ) . Moreover, lm ( f x θ + p ) = lm ( f x θ ) and lc ( f x θ + p ) = lc ( f x θ ) , so by theprevious discussion x θ f + p F −→ x θ h .(ii) Let f F −−−−→ h F −−−−→ h F −−−−→ · · · F −−−−→ h t − F −−−−→ h t : = h ; (3.1)we start with f F −→ h , if f = , then h = = p and there is nothing to prove. Let f = ,if h = then p = and hence lm ( f ) ≻ lm ( p ) ; if h = , then lm ( f ) ≻ lm ( h ) ≻ lm ( p ) ,and hence lm ( f + p ) = lm ( f ) , lc ( f + p ) = lc ( f ) , and as in the proof of the first part of (i), h + p = f + p − P ti = f i r i x α i ; but lm ( f + p ) = lm ( f ) and lc ( f + p ) = lc ( f ) , then f + p F −→ h + p .Since lm ( h i ) ≻ lm ( p ) we can repeat this reasoning for h i F −→ h i + for 1 ≤ i ≤ t −
1. Thiscomplete the proof of (ii).(iii) By (i) and using (3.1), there exists p ∈ A m with p = or lm ( f x θ ) ≻ lm ( p ) such that f x θ + p F −→ h x θ ; there exists p ∈ A m with p = ( h x θ ) ≻ lm ( p ) such that h x θ + p F −→ h x θ ; by (ii) we get that f x θ + p + p F −→ h x θ + p F −→ h x θ , i.e., p ′′ : = p + p ∈ A m is such that f x θ + p ′′ F −→ + h x θ ,with p ′′ = ( f x θ ) ≻ lm ( p ′′ ) since lm ( f x θ ) ≻ lm ( p ) and lm f ( x θ ) ≻ lm ( p ) . By inductionon t we find p ′ ∈ A m such that f x θ + p ′ F −→ + h t − x θ ,with p ′ = ( f x θ ) ≻ lm ( p ′ ) . By (i) there exists p t ∈ A m such that h t − x θ + p t F −→ hx θ ,with p t = or lm ( h t − x θ ) ≻ lm ( p t ) . By (ii), f x θ + p ′ + p t F −→ + h t − x θ + p t F −→ h x θ . Thus, f x θ + p F −→ + h x θ ,9ith p : = p ′ + p t = ( f x θ ) ≻ lm ( p ) since lm ( f x θ ) ≻ lm ( p ′ ) and lm ( f x θ ) ≻ lm ( p t ) .(iv) This is a direct consequence of (iii) taking h = . Definition 3.6.
Let A : = σ ( R ) 〈 x , . . . , x n 〉 a bijective skew P BW extension. Let θ , θ ∈ N n . We definethe following automorphism over R, ψ θ , θ : R → R that assigns to each r ∈ R. ψ θ , θ ( r ) : = σ θ + θ ( σ − θ ( r )) . Remark 3.7. (i) The inverse function of ψ is given by ψ − θ , θ ( r ) = σ θ σ − ( θ + θ ) ( r ) .(ii) Let A : = σ ( R ) 〈 x , . . . , x n 〉 a bijective skew P BW extension. For α , β , γ ∈ N n and r ∈ R , usingthe identities of Remark 1.7, we get σ β ( r ) c β , α = c β , α ψ β , α ( r ) (3.2) c β , α r = σ β ( ψ − β , α ( r )) c β , α . (3.3)Moreover, we have c β , α r c β + α , γ = σ β ( ψ − β , α ( r )) c β , α c α + β , γ = σ β ( ψ − β , α ( r )) σ β ( c α , γ ) c β , α + γ = σ β ( ψ − β , α ( r ) c α , γ ) c β , α + γ = c β , α + γ ψ β , α + γ ( ψ − β , α ( r ) c α , γ ) . (3.4)(iii) With the notation in proof of Proposition 3.5 (i); s , . . . , s t are solutions of equationlc ( h ) = t X i = lc ( f i ) c β i , α i s i ,if and only if, r i = ψ − α i , β i ( s i ) for i =
1, . . . , t , are solutions of equationlc ( h ) = t X i = lc ( f i ) c β i , α i ψ β i , α i ( r i ) .The next theorem is the theoretical support of right Division Algorithm for bijective skew P BW extensions.
Theorem 3.8.
Let F = { f , . . . , f t } be a set of non-zero vectors of A m and f ∈ A m , then the right divisionalgorithm (Algorithm 1) produces polynomials q , . . . , q t ∈ A and a reduced vector h ∈ A m w.r.t. F suchthat f F −→ + h and f = f q + · · · + f t q t + h lgorithm 1: Right division algoritm in A m Input: f , f , . . . , f t ∈ A m with f j = ( ≤ j ≤ t ) Output: q , . . . , q t ∈ A , h ∈ A m with f = f q + · · · + f t q t + h , h reduced w.r.t. { f , . . . , f t } andlm ( f ) = max { lm ( lm ( f ) lm ( q )) , . . . , lm ( lm ( f t ) lm ( q t )) , lm ( h ) } Initialization: q ← q ←
0, . . . , q t ← h ← f ; while h = and there exists j such that lm ( f j ) divides lm ( h ) do J ← { j | lm ( f j ) divides lm ( h ) } ; for i ∈ J do β j ← exp ( lm ( f j )) ; α j ← exp ( lm ( h )) − β j ; endif the equation lc ( h ) = P j ∈ J lc ( f j ) c f j , α j s j is soluble then Calculate one solution ( s j ) j ∈ J ; for j ∈ J do r j ← ψ − β j , α j ( s j ) ; q j ← q j + r j x α j ; h ← h − f j r j x α j ; endelseBreak;endend with lm ( f ) = max { lm ( lm ( f ) lm ( q )) , . . . , lm ( lm ( f t ) lm ( q t )) , lm ( h ) } .Proof. We first note that Division Algorithm is the iteration of the reduction process. If f is reducedwith respect to F : = { f , . . . , f t } , then h = f , q = · · · = q t = ( f ) = lm ( h ) . If f is not reduced,then we make the first reduction, f F −→ h , with f = P j ∈ J f j r j x α j + h , with J : = { j | lm ( f j ) divideslm ( f ) } and r j ∈ R . If h is reduced with respect to F , then the cycle While ends and we have that q j = r j x α j for j ∈ J and q j = j / ∈ J . Moreover, lm ( f ) ≻ lm ( h ) and lm ( f ) = lm ( lm ( f j ) lm ( q j )) for j ∈ J such that r j =
0, hence, lm ( f ) = max ≤ j ≤ t { lm ( lm ( f j )) lm ( q j ) , lm ( h ) } . If h is notreduced, so we make the second reduction with respect to F , h F −→ h , with h = P j ∈ J f j r j x α j + h , J : = { j | lm ( f j ) divides lm ( h ) } and r j ∈ R . We have f = P j ∈ J f j r j x α j + P j ∈ J f j r j x α j + h If h is reduced with respect to F the procedure ends and we get that q j = q j for j / ∈ J and q j = q j + r j x α j for j ∈ J . We know that lm ( f ) ≻ lm ( h ) ≻ lm ( h ) , this implies that the algorithmproduces polynomials q j with monomials ordered according to the monomial order fixed, and againwe have lm ( f ) = max ≤ j ≤ t { lm ( lm ( q j ) lm ( f j )) , lm ( h ) } . We can continue this way and the algorithmends since Mon ( A m ) is well ordered. 11 Gröbner bases for right submodules of A m In this section we present the general theory of Gröbner bases for right submodules of A m , m ≥ A = σ ( R ) 〈 x , . . . , x n 〉 is a bijective skew P BW extension of R , with R a RGS ring (see Definition3.1) and Mon ( A ) endowed with some monomial order (see Definition 2.2). A m is the right free A -module of column vectors of length m ≥
1; since A is a right Noetherian ring, then A is an I BN ring(Invariant Basis Number, see [ ] ), and hence, all bases of the free module A m have m elements. Notemoreover that A m is right Noetherian, and hence, any submodule of A m is finitely generated.The plan is to define and calculate Gröbner bases for right submodules of A m , we will give equiv-alent conditions in order to define right Gröbner bases, and finally, we will compute right Gröbnerbases using a procedure similar to right Buchberger’s algorithm over bijective skew P BW extensions.This theory was studied in the bijective general case for left modules (see [ ] ), here we adapt theseideas.Our next purpose is to define Gröbner bases for right submodules of A m . Definition 4.1.
Let M = be a right submodule of A m and let G be a non empty finite subset of non-zerovectors of M , we say that G is a Gröbner basis for M if each element = f ∈ M is reducible w.r.t. G. Wewill say that { } is a Gröbner basis for M = . Theorem 4.2.
Let M = be a right submodule of A m and let G be a finite subset of non-zero vectors ofM . Then the following conditions are equivalent: (i) G is a Gröbner basis for M . (ii)
For any vector f ∈ A m , f ∈ M if and only if f G −→ + . (iii) For any = f ∈ M there exist g , . . . , g t ∈ G such that lm ( g j ) | lm ( f ) , ≤ j ≤ t, ( i.e., ind ( lm ( g j )) = ind ( lm ( f )) and there exist α j ∈ N n such that exp ( lm ( g j )) + α j = exp ( lm ( f )) ) and lc ( f ) ∈ { lc ( g ) c g , α , . . . , lc ( g t ) c g t , α t 〉 .(iv) For α ∈ N n and ≤ v ≤ m , let { α , I 〉 v be the right ideal of R defined by { α , M 〉 v : = {{ lc ( f ) | f ∈ M , ind ( lm f ) = v , exp ( lm ( f )) = α }〉 . Then, { α , I 〉 v = J v , withJ v : = {{ lc ( g ) c g , β | g ∈ G , ind ( lm g ) = v and exp ( lm ( g )) + β = α }〉 .Proof. (i) ⇒ (ii): Let f ∈ M , if f = , then by definition f G −→ + . If f = , then there exists h ∈ A m such that f G −→ h , with lm ( f ) ≻ lm ( h ) and f − h ∈ { G 〉 ⊆ M , hence h ∈ M ; if h = , so we end.If h = , then we can repeat this reasoning for h , and since Mon ( A m ) is well ordered, we get that f G −→ + . 12onversely, if f G −→ +
0, then by the Theorem 3.8, there exist g , . . . , g t ∈ G and q , . . . , q t ∈ A such that f = g q + · · · + g t q t , i.e., f ∈ M .(ii) ⇒ (i): evident.(i) ⇔ (iii): this is a direct consequence of Definition 3.3 and the equation (3.2).(iii) ⇒ (iv) Since R is a right Noetherian ring, there exist r , . . . , r s ∈ R , f , . . . , f n ∈ M such that { α , M 〉 v = { r , . . . , r s 〉 , ind ( lm ( f i )) = v , lm ( f i ) = x α , 1 ≤ i ≤ n , with { r , . . . , r s 〉 ⊆ { lc ( f ) , . . . , lc ( f n ) 〉 ,then { lc ( f ) , . . . , lc ( f n ) 〉 = { α , M 〉 v . Let r ∈ { α , M 〉 v , there exist a , . . . , a n ∈ R such that r = lc ( f ) a + · · · + lc ( f n ) a n ; by (iii), for each i there exist g i , . . . , g t i i ∈ G and b ji ∈ R such thatlc ( f i ) = lc ( g i ) c g i , α i b i + · · · + lc ( g t i i ) c g tii , α tii b t i i ,with v = ind ( lm f i ) = ind ( lm ( g ji )) and α = exp ( lm ( f i )) = α ji + exp ( lm ( g ji )) , thus { α , M 〉 v ⊆ J v .Conversely, if r ∈ J v , then r = lc ( g ) c g , β b + · · · + lc ( g t ) c g t , β t b t , with b i ∈ R , β i ∈ N n , g i ∈ G such that ind ( lm ( g i )) = v , β i + exp ( lm ( g i )) = α for any 1 ≤ i ≤ t .Note that g i x β i ∈ M , ind ( g i x β i ) = v , exp ( lm ( g i x β i )) = α , lc ( g i x β i ) = lc ( g i ) c g i , β i , for 1 ≤ i ≤ t ,thus r = lc ( g x β ) b + · · · + lc ( g t x β t ) b t , i.e., r ∈ { α , M 〉 v .(iv) ⇒ (iii): let = f ∈ M and let α = exp ( lm ( f )) and v = ind ( lm ( f )) , then lc ( f ) ∈ { α , M 〉 v ; by(iv) lc ( f ) = lc ( g ) c g , β b + · · · + lc ( g t ) c g t , β t b t , with b i ∈ R , β i ∈ N n , g i ∈ G such that = ind ( lm ( g i )) = v and β i + exp ( lm ( g i )) = α for any 1 ≤ i ≤ t . From this we conclude that, lm ( g i ) | lm ( f ) .Like consequence of this theorem we get the following results. Corollary 4.3.
Let M = be a right submodule of A m . Then, (i) If G is a Gröbner basis for M , then M = 〈 G 〉 . (ii) Let G be a Gröbner basis for M , if f ∈ M and f G −→ + h , with h reduced, then h = . (iii) Let G = { g , . . . , g t } be a set of non-zero vectors of M with lc ( g i ) = for each ≤ i ≤ t , if given = r ∈ M there exists i such that lm ( g i ) | lm ( r ) . Then, G is a Gröbner basis of M . Proof. (i) This is a direct consequence of Theorem 4.2.(ii) Let f ∈ M and f G −→ + h , with h reduced; since f − h ∈ 〈 G 〉 = M , then h ∈ M ; if h = then h can be reduced by G , but this is not possible since h is reduced.(iii) let f ∈ A m , by Theorem 3.8 there exists r reduced such that f G −→ + r . If f ∈ M then r ∈ M ;if r = , then by hypothesis there exists g i ∈ G such that lm ( g i ) divides lm ( r ) , thus, since lc ( g i ) =
1, then r is reducible, which is a contradiction and therefore, f G −→ + . On the otherhand, if f G −→ + , then f ∈ M . From Theorem 4.2 (ii), we get that G is Gröbner basis of M . Corollary 4.4.
Let G be a Gröbner basis for a right submodule M of A m . Given g ∈ G, if g is reduciblew.r.t. G ′ = G − { g } , then G ′ is a Gröbner basis for M . roof. According to Theorem 4.2, it is enough to show that every f ∈ M is reducible w.r.t. G ′ . Let f be a nonzero vector in M ; since G is a Gröbner basis for M , f is reducible w.r.t. G and there existelements g , . . . , g t ∈ G satisfying the conditions (i), (ii) and (iii) in the Definition 3.3. If g = g i foreach 1 ≤ i ≤ t , then we finished. Suppose that g = g j for some j ∈ {
1, . . . , t } and let β i = exp ( lm ( g i )) for i = j , β = exp ( lm ( g )) , and α i , α ∈ N n such that α i + β i = exp ( lm ( f )) = α + β . Thus,lc ( f ) = lc ( g ) c β , α r + · · · + lc ( g ) c β , α r j + · · · + lc ( g t ) c β t , α t r t .On the other hand, since g is reducible w.r.t. G ′ , there exist g ′ , . . . , g ′ s ∈ G ′ such that lm ( g ′ l ) | lm ( g ) and lc ( g ) = P sk = lc ( g ′ k ) c β ′ k , α ′ k r ′ k , where β ′ k = exp ( lm ( g ′ k )) , α ′ k ∈ N n and α ′ k + β ′ k = exp ( lm ( g )) = β .Thus, lm ( g ′ k ) | lm ( f ) for 1 ≤ i ≤ s ; moreover, using the equation (3.4) of Remark 3.7, we have that c β ′ k , α ′ k r ′ k c β , α = c β ′ k , α ′ k r ′ k c β ′ k + α ′ k , α = c β ′ k , α ′ k + α r ′′ k ,where r ′′ k = ψ β ′ k , α ′ k + α ( ψ − β ′ k , α ′ k ( r ) c α ′ k , α ) . Therefore,lc ( g ) c β , α = s X k = lc ( g ′ k ) c β ′ k , α ′ k r ′ k c β , α = s X k = lc ( g ′ k ) c β ′ k , α ′ k + α r ′′ k .Since α + β = exp ( lm ( f )) , then α + α ′ k + β ′ k = exp ( lm ( f )) . Further, if exists g w ∈ { g , . . . , g t } suchthat g w = g ′ z for some z ∈ {
1, . . . , s } , then β w = β ′ z and α + α ′ z = α w ; therefore, in the representationof lc ( f ) would appear the term lc ( g w ) c β w , α w ( r w + r ′′ z r j ) . From this we get that f is reducible w.r.t. G ′ and, hence, G ′ is a Gröbner basis for M . Recall that we are assuming that A is a bijective skew P BW extension, we will prove in the presentsection that every submodule M of A m has a Gröbner basis, and also we will construct the Buch-berger’s algorithm for computing such bases.We start fixing some notation and proving a preliminary general result. Definition 5.1.
Let F : = { g , . . . , g s } ⊆ A m , X F the least common multiple of { lm ( g ) , . . . , lm ( g s ) } , θ ∈ N n , β i : = exp ( lm ( g i )) and γ i ∈ N n such that γ i + β i = exp ( X F ) , ≤ i ≤ s. B F , θ will denote a finiteset of generators in R s of right R-moduleS F , θ : = Syz rR [ lc ( g ) c β , γ + θ · · · lc ( g s ) c β s , γ s + θ ] .For θ = : = (
0, . . . , 0 ) , S F , θ will be denoted by S F and B F , θ by B F . Remark 5.2.
Let ( b , . . . , b s ) ∈ S F , θ . Since A is bijective, then there exists an unique ( b ′ , . . . , b ′ s ) ∈ S F such that b i = ψ β i , γ i + θ ( ψ − β i , γ i ( b ′ i ) c γ i , θ ) , for each 1 ≤ i ≤ s .In fact, the existence and uniqueness of ( b ′ , . . . , b ′ s ) it follows from the bijectivity of A . Now, since ( b , . . . , b s ) ∈ S F , θ , then P si = lc ( g i ) c β i , γ i + θ b i =
0. Replacing b i in the last equation, we obtain s X i = lc ( g i ) c β i , γ i + θ ψ β i , γ i + θ ( ψ − β i , γ i ( b ′ i ) c γ i , θ ) = c β i , γ i + θ ψ β i , γ i + θ ( ψ − β i , γ i ( b ′ i ) c γ i , θ ) = c β i , γ i b ′ i c β i + γ i , θ ,thus, P si = lc ( g i ) c β i , γ i b ′ i c β i + γ i , θ =
0, and since c β i + γ i , θ = c X F , θ is invertible, then P si = lc ( g i ) c β i , γ i b ′ i =
0, i.e., ( b ′ , . . . , b ′ s ) ∈ S F . Lemma 5.3.
Let g , . . . , g s ∈ A , c , . . . , c s ∈ R − { } and α , . . . , α s , β , . . . , β s ∈ N n such that, X δ : = lm ( lm ( g i ) x α i ) and β i : = exp ( g i ) , for each ≤ i ≤ s. If lm ( P si = g i c i x α i ) ≺ X δ , then there existr , . . . , r k ∈ R and z , . . . , z s ∈ A such that s X i = g i c i x α i = k X j = (cid:129) s X i = g i ψ − β i , γ i ( b ji ) x γ i ‹ r j x δ − exp ( X F ) + s X i = g i z i , where X F is the least common multiple of lm ( g ) , . . . , lm ( g s ) , γ i ∈ N n is such that γ i + β i = exp ( X F ) ,for each ≤ i ≤ s, and B F = { b , . . . , b k } : = { ( b , . . . , b s ) , . . . , ( b k , . . . , b ks ) } . Moreover, lm ( P si = g i ψ − β i , γ i ( b ji ) x γ i r j x δ − exp ( X F ) ) ≺ X δ for every ≤ j ≤ k, and lm ( g i z i ) ≺ X δ for every ≤ i ≤ s.Proof. Since X δ = lm ( lm ( g i ) x α i ) , then lm ( g i ) | X δ and hence X F | X δ , so there exists θ ∈ N n suchthat exp ( X F ) + θ = δ . On the other hand, γ i + β i = exp ( X F ) and α i + β i = δ , so α i = γ i + θ foreach 1 ≤ i ≤ s . Now, lm ( P si = g i c i x α i ) ≺ X δ implies that P si = lc ( g i ) σ β i ( c i ) c β i , α i =
0. From equation(3.2) of Remark 3.7, we have P si = lc ( g i ) c β i , α i d i =
0, with d i = ψ β i , α i ( c i ) , for each 1 ≤ i ≤ s . So, P si = lc ( g i ) c β i , γ i + θ d i =
0. This implies that ( d , . . . , d s ) ∈ S F , θ ; from Remark 5.2 we know that existsa unique ( d ′ , . . . , d ′ s ) ∈ S F such that d i = ψ β i , γ i + θ ( ψ − β i , γ i ( d ′ i ) c γ i , θ ) . Then, c i = ψ − β i , α i ( d i ) = ψ − β i , γ i + θ ( d i ) = ψ − β i , γ i ( d ′ i ) c γ i , θ ,therefore, we have s X i = g i c i x α i = s X i = g i ψ − β i , γ i ( d ′ i ) c γ i , θ x α i = s X i = g i ψ − β i , γ i ( d ′ i ) c γ i , θ x γ i + θ = s X i = g i ψ − β i , γ i ( d ′ i )( x γ i x θ − p γ i , θ )= s X i = g i ψ − β i , γ i ( d ′ i ) x γ i x θ + s X i = g i p i with p i = ( p i ) ≺ x θ + γ i , hence, g i p i = ( g i p i ) ≺ x θ + γ i + β i = X δ . On the other hand,since ( d ′ , . . . , d ′ s ) ∈ S F , then there exist r ′ , . . . , r ′ k ∈ R such that ( d ′ , . . . , d ′ s ) = b r ′ + · · · + b k r ′ k = b , . . . , b s ) r ′ + · · · + ( b k , . . . , b ks ) r ′ k , thus d ′ i = P kj = b ji r ′ j . Therefore, we have s X i = g i ψ − β i , γ i ( d ′ i ) x γ i x θ = s X i = g i ψ − β i , γ i € k X j = b ji r ′ j Š x γ i x θ = s X i = g i € k X j = ψ − β i , γ i ( b ji ) ψ − β i , γ i ( r ′ j ) Š x γ i x θ = s X i = g i € k X j = ψ − β i , γ i ( b ji )( x γ i σ − γ i ( ψ − β i , γ i ( r ′ j )) + q ′ i j ) Š x θ ,with q ′ i j = ( q ′ i j ) ≺ x γ i , we note that ψ − β i , γ i ( r ) = σ γ i ( σ − ( γ i + β i ) ( r )) = σ γ i ( σ − exp ( X F ) ( r )) , hencewe have s X i = g i ψ − β i , γ i ( d ′ i ) x γ i x θ = s X i = g i € k X j = ψ − β i , γ i ( b ji )( x γ i σ − exp ( X F ) ( r ′ j ) + q ′ i j ) Š x θ = k X j = s X i = g i ψ − β i , γ i ( b ji ) x γ i r j x θ + s X i = k X j = g i q i j x θ = k X j = € s X i = g i ψ − β i , γ i ( b ji ) x γ i Š r j x θ + s X i = g i q i ,where q i j : = ψ − β i , γ i ( b ji ) q ′ i j , r j : = σ − exp ( X F ) ( r ′ j ) and so, q i : = P kj = q i j x θ = ( q i ) ≺ x θ + γ i .Finally we get, s X i = g i c i x α i = k X j = (cid:129) s X i = g i ψ − β i , γ i ( b ji ) x γ i ‹ r j x δ − exp ( X F ) + s X i = g i z i ,with z i : = p i + q i for 1 ≤ i ≤ s . Is easy to see that lm ( P si = g i ψ − β i , γ i ( b ji ) x γ i r j x θ ) ≺ X δ since thatlm ( P si = g i ψ − β i , γ i ( b ji ) x γ i ) ≺ x γ i + β i , and lm ( g i z i ) = lm ( g i p i + g i q i ) ≺ X δ .With the notation of Definition 5.1 and Lemma 5.3, we will prove the main result of the presentsection. Theorem 5.4.
Let M = be a right submodule of A m and let G be a finite subset of non-zero generatorsof M . Then the following conditions are equivalent. (i) G is a Gröbner basis of M . (ii)
For all F : = { g , . . . , g s } ⊆ G, with X F = and for any ( b , . . . , b s ) ∈ B F , s X i = g i ψ − β i , γ i ( b i ) x γ i G −→ + .16 roof. (i) ⇒ (ii): We observe that f : = P si = g i ψ − β i , γ i ( b i ) x γ i ∈ M , so by Theorem 4.2 f G −→ + .(ii) ⇒ (i): Let = f ∈ M , we will prove that the condition (iii) of Theorem 4.2 holds. Let G : = { g , . . . , g t } , then there exist h , . . . , h t ∈ A such that f = g h + · · · + g t h t and we can choose { h i } ti = such that X δ : = max { lm ( lm ( g i ) lm ( h i )) } ti = is minimal. Let lm ( h i ) : = x α i , c i : = lc ( h i ) , lm ( g i ) : = x β i for 1 ≤ i ≤ t and F : = { g i ∈ G | lm ( lm ( g i ) lm ( h i )) = X δ } ; renumbering the elements of G we canassume that F = { g , . . . , g s } . We will consider two possible cases. Case 1 : lm ( f ) = X δ . Then lm ( g i ) | lm ( f ) for 1 ≤ i ≤ s andlc ( f ) = s X i = lc ( g i ) σ β i ( c i ) c β i , α i = s X i = lc ( g i ) c β i , α i ψ β i , α i ( c i ) .i.e., the condition (iii) of Theorem 4.2 holds. Case 2 : lm ( f ) ≺ X δ . We will prove that this produces a contradiction. To begin, note that f canbe written as f = s X i = g i c i x α i + s X i = g i ( h i − c i x α i ) + t X i = s + g i h i ; (5.1)we have lm ( g i ( h i − c i x α i )) ≺ X δ for each 1 ≤ i ≤ s and lm ( g i h i ) ≺ X δ for each s + ≤ i ≤ t , solm ( s X i = g i ( h i − c i x α i )) ≺ X δ and lm ( t X i = s + g i h i ) ≺ X δ ,and hence lm ( P si = g i c i x α i ) ≺ X δ . By Lemma 5.3 (and its notation), we have s X i = g i c i x α i = k X j = € s X i = g i ψ − β i , γ i ( b ji ) x γ i Š r j x δ − exp ( X F ) + s X i = g i z i , (5.2)where lm ( P si = g i ψ − β i , γ i ( b ji ) x γ i x δ − exp ( X F ) ) ≺ X δ for each 1 ≤ j ≤ k and lm ( g i z i ) ≺ X δ for 1 ≤ i ≤ s .By the hypothesis, P si = g i ψ − β i , γ i ( b ji ) x γ i G −→ + , whence, by Theorem 3.8, there exist q , . . . , q t ∈ A such that s X i = g i ψ − β i , γ i ( b ji ) x γ i = t X i = g i q i ,with lm ( P si = g i ψ − β i , γ i ( b ji ) x γ i ) = max { lm ( lm ( g i ) lm ( q i )) } ti = , but ( b j , . . . , b js ) ∈ B F , thus using theequation (3.3) of Remark 3.7, we getlc ( s X i = g i ψ − β i , γ i ( b ji ) x γ i ) = s X i = lc ( g i ) σ β i ( ψ − β i , γ i ( b ji )) c β i , γ i = s X i = lc ( g i ) c β i , γ i b ji = €P si = g i ψ − β i , γ i ( b ji ) x γ i Š ≺ X F and hence lm ( lm ( g i ) lm ( q i )) ≺ X F for each 1 ≤ i ≤ t . Therefore, k X j = € s X i = g i ψ − β i , γ i ( b ji ) x γ i Š r j x δ − exp ( X F ) = k X j = € t X i = g i q i Š r j x δ − exp ( X F ) t X i = k X j = g i q i r j x δ − exp ( X F ) = t X i = g i e q i ,with e q i : = P kj = q i r j x δ − exp ( X F ) and lm ( g i e q i ) ≺ X δ for each 1 ≤ i ≤ t . Substituting P si = g i c i x α i = P ti = g i e q i + P si = g i z i into equation (5.1), we obtain f = t X i = g i e q i + s X i = g i ( h i − c i x α i ) + s X i = g i z i + t X i = s + g i h i ,and so we have expressed f as a combination of vectors g , . . . , g t , where every term has leadingmonomial ≺ X δ . This contradicts the minimality of X δ and we finish the proof. Corollary 5.5.
Let F = { f , . . . , f s } be a set of non-zero vectors of A m . The algorithm below (Algorithm2) produces a Gröbner basis for the right submodule 〈 F 〉 of A m ( P ( X ) denotes the set of subsets of theset X ) . Algorithm 2:
Right Buchberger’s algorithm in A m Input: F : = { f , . . . , f s } ⊆ A m , f i = , 1 ≤ i ≤ s Output: G = { g , . . . , g t } a Gröbner basis for 〈 F 〉 Initialization: G ← ; , G ′ ← F ; while G ′ = G do D ← P ( G ′ ) − P ( G ) ; G ← G ′ ; for each S : = { g i , . . . , g i k } ∈ D with X S = Compute B S ; for each b = ( b , . . . , b k ) ∈ B S do Reduce P kj = g i j ψ − β ij , γ j ( b j ) x γ j G ′ −−→ + r ; with r reduced w.r.t.. G ′ ; β i j , γ j as in Def 5.1 if r = G ′ ← G ′ ∪ { r } ; endendendend From Theorem 5.4 and the previous corollary we get the following direct conclusion.
Corollary 5.6.
Each right submodule of A m has a Gröbner basis. Examples implemented in
SPBWE.lib library
The extensions skew
P BW extensions were implemented in Maple using the
SPBWE.lib library (see [ ] ), which allows to make basic calculations with this type of rings and can also provide answersto several homological problems such as the computation of sizigias, within the library are alreadydeveloped the algorithms that we present in this paper: the right division algorithm and the rightBuchberger algorithm, below here we will present only a brief view of its execution. Example 6.1.
Consider the diffusion algebra A : = σ ( Q [ x , x ]) 〈 D , D 〉 subject to relation: D D = D D + x D − x D .Taking the following polynomials in Af : = x x D D + x x D : f : = x x D D : f : = x D : f : = x D :We use the right division algorithm over these polynomials as follow > DivisionAlgorithm ( f , [ f , f , f ] , gradlexrev, A , right ) and the value returned is x D + x x − x x D x x , 0 Therefore we get that x x D D + x x D ∈ 〈 x x D D , x D , x D 〉 A with x x D D + x x D = x x D D ( x D + x x ) + x D € − x x D Š + x D ( x x ) .The following example is a non trivial instance of applicability of the SPBWE.lib library, inparticular, is possible to define iterated skew
P BW extensions in the library and compute left or rightGröbner bases over theses.
Example 6.2.
Let A = R [ w , ϕ ] the skew polynomial ring of endomorphism type with R = C and ϕ : C → C , such that ϕ ( q ) = q , for q ∈ C . Using the SPBWE.lib we can to define the extension C = σ ( A ) 〈 x , y , z 〉 , subjects to relations y x = p x y , z x = wx z , z y = w yz ,with endomorphims σ , σ , σ : A → A , making σ ( w ) = w , σ ( w ) = w , σ ( w ) = − w , and σ i -derivations δ i null for i =
1, 2, 3.Now a instance of right division algorithm and other of Buchberger algorithm over C .19. Considering the right ideal J : = { i − iwz , 3 iw y − iwx , y , 1 〉 C , now we verify if r : = wx y + wx + ( + i ) wz − ∈ J , for this purpose we make in Maple as follow > J : = [ I − I · w · z , 3 I · w · y − I · w · x , y , 1 ] > DivisionAlgorithm( w · x · y + w · x + ( + I ) w · z − J , gradlex, C , right) we get − − I − I y − I w y + w − + I , 0 Thus, r is in ideal J .2. As instances of right Buchberger algorithm on the extension C . Consider the right C -submodule M : = 〈 ( w , x + w y ) , ( − z ) , ( w yz , 4 z + y ) 〉 C in C . Using the following statements in Maple > w x + w y − zw yz z + y > BuchbergerAlgSkewPoly( M , gradlex, TOP, C , right); we get a right Gröbner basis of M (cid:20)(cid:20) w x + w y (cid:21) , (cid:20) − z (cid:21) , (cid:20) wx y z + y (cid:21) , (cid:20) w z (cid:21) , (cid:20) − w y (cid:21) , (cid:20) pw y (cid:21)(cid:21) As works to be developed directly as a consequence of the results obtained, it is already possible toconstruct and effectively implement algorithms over bijective skew
P BW extensions to answer prob-lems of noncommutative algebraic geometry, algebraic analysis and particulary to compute syzygiesof ideals and modules over this type of extensions, and with it to be able to give answer to variousproblems of the homological algebra as computation of right inverse matrices, the computation ofExt homology groups and homological questions arising from the computation of Ext. With this newpossibility of algorithms the idea is to complete the
SPBWE.lib library and provide support on prob-lems in areas of noncommutative algebra that have been never implemented computationally andsolved before.