Graded polynomial identities and central polynomials of matrices over an infinite integral domain
aa r X i v : . [ m a t h . R A ] A ug GRADED POLYNOMIAL IDENTITIES AND CENTRALPOLYNOMIALS OF MATRICES OVER AN INFINITEINTEGRAL DOMAIN
LU´IS FELIPE GONC¸ ALVES FONSECA
Abstract.
Let K be an infinite integral domain and M n ( K ) be the algebraof all n × n matrices over K . This paper aims for the following goals: • Find a basis for the graded identities for elementary grading in M n ( K )when the neutral component and diagonal coincide; • Describe the Z p -graded central polynomials of M p ( K ) when p is a primenumber; • Describe the Z -graded central polynomials of M n ( K ). Introduction
Polynomial Identity theory (PI) is an important branch of the Ring Theory.The first crucial developments in PI-theory were Kaplansky’s Theorem [14] aboutprimitive PI-algebras and the Amitsur-Levitsky Theorem [1], published in 1948 and1950, respectively. The latter theorem is important for describing the polynomialidentities of matrices.Let K be a field and M n ( K ) be the algebra of all n × n matrices over K . PI-theory is used to obtain a basis for polynomial identities of M n ( K ). Razmyslov[21] discovered a nine-polynomial basis for the identities of M ( K ) when K is afield of characteristic zero. Some years later, Drensky [11] found a minimal poly-nomial basis: comprising the Hall identity and the standard polynomial of degree4. Koshlukov [18] found a basis (consisting of four identities) for the identities of M ( K ) when K is an infinite field of charK = p > M ( K ) when K is an infinite field ofcharacteristic 2 or an infinite integral domain remain unresolved.Let K be a field of characteristic zero. In 1950 Specht [24] conjectured that everysystem of identities in associative algebra has a finite basis. Specht’s conjecture wasunsolved until the late 1980s when Kemer demonstrated its truth using the theoryof Z -graded algebras.Another important problem is describing the graded polynomial identities of M n ( K ). The Z -graded polynomial identities of M ( K ) were described by Di Vin-cenzo [10], while Vasilovsky [25] described the Z n -graded polynomial identities of M n ( K ). A year earlier, the same author had described the Z -graded identities of M n ( K ) [26]. Vasilovsky’s results were extended by Bahturin and Drensky [4], whofound the basis of the graded identities for the elementary gradings on M n ( K ) whenthe neutral component and diagonal of M n ( K ) coincide. Azevedo [2],[3] and Silva[23] extended the Vasilovsky’s and Bahturin-Drensky’s results, respectively to aninfinite field. Kaplansky [15] posed a list of open problems in Ring Theory. Among these wasthe question does a non-trivial contral polynomial exist for M n ( K ) when n ≥ M n ( K ) is a crucial task in PI-theory.When K is a field of characteristic zero, a set of generators may be found forthe central polynomials of M ( K ) [19]. Koshlukov and Colombo [9] described thecentral polynomials of M ( K ), when K is an infinite field of characteristic p > M n ( K ) were made byBrand˜ao J´unior [8]. Assuming an infinite ground field K , he described the Z n -graded central polynomials of M n ( K ) when charK ∤ n , as well as, the Z p -gradedcentral polynomials of M p ( K ) (where charK = p >
2) and the Z -graded centralpolynomials of M n ( K ).Few reports of graded identities of M n ( K ) exist in the literature, when K is aninfinite integral domain. Brand˜ao J´unior, Koshlukov and Krasilnikov [7] detaileda basis for the Z -graded identities of M ( K ). They also described a basis for the Z -graded central polynomials of M ( K ).In this paper, we combine the methods of [2], [3], [4], [8], [12], [23], [25] and [26].This paper aims for the following goals: • Find a basis for the graded identities for elementary grading in M n ( K )when the neutral component and diagonal coincide; • Describe the Z p -graded central polynomials of M p ( K ) when p is a primenumber; • Describe the Z -graded central polynomials of M n ( K ).In these three situations, K is an infinite integral domain.2. Preliminaries
Let K be a fixed unital associative and commutative ring . We assume that allmodules are left-modules over K and all (unital associative) algebras are consideredover K . We assume that all ideals are bilateral ideals. The set { , · · · , n } is denotedby b n . Moreover, G denotes an arbitrary group and N = { , , , · · · , n, · · · } is theset of natural numbers. Here, S n denotes the group of permutations on b n . H n denotes the subgroup of S n generated by (12 · · · n ).Let X be a countable set of variables and K h X i be the free associative ringfreely generated by X . Let A be an algebra over K and let Z ( A ) be the cen-ter of A . A polynomial f ( x , · · · , x n ) ∈ K h X i is called an ordinary polynomialidentity (respectively an ordinary central polynomial) for A if f ( a , · · · , a n ) = 0for all a , · · · , a n ∈ A (respectively f (0 , · · · ,
0) = 0 and f ( a , · · · , a n ) ∈ Z ( A )).The algebra A is a PI-algebra if there exists f ∈ K h X i satisfying the followingconditions: : f is an ordinary polynomial identity for A ; : Some coefficient in the highest-degree homogeneous component of f equalsto 1.The set of ordinary polynomial identities (respectively ordinary central polyno-mials) of A is an ideal (respectively submodule) of K h X i that is invariant underall endomorphisms of K h X i . The ideals (respectively submodules) of K h X i that ATRICES OVER AN INFINITE INTEGRAL DOMAIN 3 are invariant under all endomorphisms of K h X i are called T -ideals (respectively T -spaces).Clearly, the intersection of a family of T -ideals (respectively T -spaces) of K h X i is also a T -ideal (respectively a T -space). Let S be a non-empty set of K h X i . The T -ideal (respectively T -space) generated by S , denoted h S i T (respectively h S i T ), isthe intersection of all T -ideals (respectively T -spaces) containing S .An algebra A over K is G -graded when there exist K -submodules { A g } g ∈ G ⊂ A such that: A = L g ∈ G A g (1); A g A h ⊂ A gh for all g, h ∈ G (2).Each submodule A g is called the homogeneous component of G -degree g andits non-zero elements are homogeneous elements of G -degree g . Moreover, if a isa homogeneous element, its G -degree is denoted by α ( a ). We denote the identityelement of G by e . The support of A , with respect to the grading { A g } g ∈ G , is thefollowing subset of G : Supp G ( A ) := { g ∈ G | A g = { }} .Let { X g | g ∈ G } be a family of disjoint countable sets indexed by G and let X = S g ∈ G X g . K h X i g is the K -submodule of K h X i spanned by m = x j · · · x j k such that α ( m ) = g . The decomposition K h X i = L g ∈ G K h X i g is a G -grading,whereby K h X i is the G -graded free associative ring freely generated by X . Amonomial is a variable or a product of variables in X .Let m = x i · · · x i l be a monomial of K h X i . We denote by h ( m ) the l -tuple( α ( x i ) , · · · , α ( x i l )).An endomorphism φ of K h X i is called G -graded endomorphism when φ ( K h X i g ) ⊂ K h X i g ∀ g ∈ G . When a graded ideal (respectively a graded submodule) I ⊂ K h X i is invariant under all G -graded endomorphisms of K h X i is called a T G -ideal (re-spectively a T G -space). A graded polynomial f ( x , · · · , x n ) ∈ K h X i is a G -graded polynomial identity for A (respectively a G -graded central polynomial for A ) if f ( a , · · · , a n ) = 0 for all a i ∈ A α ( x i ) , i = 1 , · · · , n ( f (0 , · · · ,
0) = 0 and f ( a , · · · , a n ) ∈ Z ( A ) for all a i ∈ A α ( x i ) , i = 1 , · · · , n ).The set of all G -graded identities of A (respectively all G -graded central polyno-mials of A ) is denoted by T G ( A ) (respectively C G ( A )). Clealy, T G ( A ) (respectively C G ( A )) is a T G -ideal (respectively a T G -space and a subalgebra) and the intersec-tion of a family of T G -ideals (respectively T G -spaces) of K h X i is also a T G -ideal(respectively a T G -space). The T G -ideal (respectively T G -space) generated by a non-empty set S of K h X i is defined as in the ordinary case. The T G -ideal (respectivelya T G -space) generated by S is denoted by h S i T G (respectively h S i T G ). A gradedpolynomial f is said to be a consequence of S ⊂ K h X i if f ∈ h S i T G (or equivalently,that f follows from S ). A set S ⊂ K h X i is called a basis for the graded identities(respectively the graded central polynomials) of A if T G ( A ) = h S i T G (respectively C G ( A ) = h S i T G ).The matrix unit e ij ∈ M n ( K ) contains 1 as the only a non-zero value in the i -th row and j -th column. Given an n -tuple g = ( g , · · · , g n ) ∈ G n , a G -gradingis determined in M n ( K ) by stipulating that e ij is homogeneous of G -degree g − i g j .These gradings are elementary and we say that M n ( K ) possesses an elementarygrading induced by g . We equipped M n ( K ) with an elementary grading inducedby an n -tuple of distinct elements from G . The set { g , · · · , g n } is denoted by G n . LU´IS FELIPE GONC¸ ALVES FONSECA
Below we provided two important examples of elementary gradings on M n ( K )whose neutral component and diagonal coincide: : Z n -canonical grading (or Z n -grading): when G = Z n and the n -tuple g is(1 , , · · · , n − , n ); : Z -canonical grading (or Z -grading): when G = Z and the n -tuple g is(1 , , · · · , n − , n ).3. Silva’s Generic Model
Generic models have an important role in PI (see for instance [5], [6] and [20]).In this section, we recall Silva’s Generic Model, as described in [23]. Let G be anarbitrary group and let g = ( g , · · · , g n ) ∈ G n be an n -tuple of distinct elementsfrom G . We consider the algebra M n ( K ) to be equipped with the elementarygrading induced by g . For each h ∈ G , let Y h = { y kh,i | ≤ k ≤ n ; i ≥ } be acountable set of commuting variables and let Y = S h ∈ G Y h . Let Ω = K [ Y ] denotethe polynomial ring with commuting variables in Y . Let M n (Ω) be the algebraof all n × n matrices over Ω. This algebra may be equipped with the elementarygrading induced by g as M n ( K ). Let G n denote the set { g , · · · , g n } . Definition 3.1.
Let h be an element of G . Let L h denote, the set of all indices k ∈ b n such that g k h ∈ G n . Let s kh denote the index determined by g s kh := g k h .Let h = ( h , · · · , h m ) ∈ G m . L h defines (the set of indices associated with the m -tuple ( h , · · · , h m )) the subset of b m such that its elements satisfy the followingproperty: g k h · · · h i ∈ G n ∀ i ∈ b m .We define a sequence ( s k , · · · , s km +1 ) (the sequence associated with h is deter-mined by k ) inductively by setting: : s k = k ; : s kl : g s kl = g k h · · · h l − ∀ l ∈ { , · · · , m + 1 } A generic matrix of G -degree h is a homogeneous element of M n (Ω) the followingtype: A hi = P k ∈ L h y kh,i e k,s kh .The G -graded subalgebra of M n (Ω) generated by the generic matrices is called thealgebra of the generic matrices which we denote by R . Notice that Supp G ( R ) = Supp G ( M n ( K )).The next lemma is an important computational result. Its proof is the immediateconsequence of a multiplication table of matrix units. Lemma 3.2. ( [23] , Lemma 3.5) If L is the set of indices associated with the q -tuple ( h , · · · , h q ) in G q and s k = ( s k , · · · , s kq +1 ) denotes the corresponding sequencedetermined by k ∈ L ,then A h i · · · A h q i q = P k ∈ L w k e s k ,s kq +1 which w k = y s k h ,i y s k h ,i · · · y s kq h q ,i q . Definition 3.3.
Let f ( x , · · · , x n ) be a polynomial of K h X i and let A ∈ R α ( x ) , · · · ,A n ∈ R α ( x n ) . f ( A , · · · , A n ) denotes the result of replacing for the correspondingelements of R . ATRICES OVER AN INFINITE INTEGRAL DOMAIN 5
The next lemma is the same in ([23], Lemma 4.5).
Lemma 3.4.
Let M ( x , · · · , x q ) and N ( x , · · · , x q ) be two monomials of K h X i that start with the same variable. Let m ( x , · · · , x q ) , n ( x , · · · , x q ) be two mono-mials obtained from M and N respectively by deleting the first variable. If thereexist matrices A , · · · , A q , such that M ( A , · · · , A q ) and N ( A , · · · , A q ) have, inthe same position, the same non-zero entry ,then the matrices m ( A , · · · , A q ) and n ( A , · · · , A q ) also have, in the same position, the same non-zero entry.Proof. It is the immediate consequence of Lemma 3.2. (cid:3)
Remark 3.5. ( [23] , Corollary 3.7) Notice that if m and m are two monomialssuch that h ( m ) = h ( m ) , then m ∈ T G ( R ) if and only if m ∈ T G ( R ) . Remark 3.6.
Let m be a monomial. Notice that m ∈ T G ( R ) if and only if m ∈ T G ( M n ( K )) . The next lemmas three lemmas can be proved by elementary algebraic methods.
Lemma 3.7.
Let f ∈ T G ( R ) . Then all multi-homogeneous components of f areelements of T G ( R ) . Lemma 3.8.
Let f ∈ C G ( R ) . Then all multi-homogeneous components of f areelements of C G ( R ) . Lemma 3.9.
Let m ( x , · · · , x q ) = x i · · · x i r and n ( x , · · · , x q ) be two monomialssuch that the matrices n ( A , · · · , A q ) and m ( A , · · · , A q ) have, at some position,the same non-zero entry. Then m − n is a multi-homogeneous polynomial. Observe that T G ( R ) ⊂ T G ( M n ( K )). The proof of the next lemma is similar tothat proof in ([2],Lemma 3). Lemma 3.10.
Let K be an infinite integral domain.Then T G ( M n ( K )) = T G ( R ) . Corollary 3.11.
Let K be an infinite integral domain.Then C G ( M n ( K )) = C G ( R ) . Some graded identities of R and Type 1-monomials In this section, we present some graded identities for elementary grading in R when the neutral component and diagonal coincide. Notice that R is equipped withthe elementary grading induced by an n -tuple of pairwise elements from G if andonly if R e and diagonal coincide.The next lemma was proved by Bahturin and Drensky in ([4], Lemma 4.1) forfull algebra of n by n matrices over a field of characteristic zero. Lemma 4.1.
The following graded polynomials are G -graded polynomial identitiesfor R : • x x − x x when α ( x ) = α ( x ) = e (1) ; • x x x − x x x when α ( x ) = α ( x ) = ( α ( x )) − = e (2) ; • x when R α ( x ) = { } (3) .Proof. It follows from Lemma 3.2 and the proof of Lemma 4.1 in [4]. (cid:3)
Definition 4.2.
Let J be the T G -ideal generated by (1) , (2) and (3) . Let J be the T G -ideal generated by (1) and (2) only. LU´IS FELIPE GONC¸ ALVES FONSECA
Definition 4.3.
Let m = x i · · · x i q be a G -graded monomial. An element n ∈ K h X i is called a subword of m when there exist j ∈ { , · · · , q } and l ∈ N , where j + l ≤ q , such that: n = x i j · · · x i j + l .Likewise, the monomial n is termed a proper subword of m when n is a subword of m , n = m and n = 1 . Definition 4.4.
Let m = x i · · · x i l be a G -graded monomial. This monomial iscalled a Type 1-monomial when the G -degree of all its non-empty subwords areelements of Supp G ( R ) . ([4]) shows an example of G -graded Type 1-monomial identities of M n ( K ) (seeExample 4.7,[4]).In this paper, the following lemma is useful. Lemma 4.5.
Let m be a multilinear Type 1-monomial. Let m be the polynomialobtained from m by deleting the variables of G -degree e by one. Then m ∈ T G ( R ) if and only if m ∈ T G ( R ) . The proof of the following lemma is left as an exercise.
Lemma 4.6.
Let m ∈ T G ( R ) be a monomial. If m is not a Type 1-monomialidentity, then m follows from (3) . Type 1-monomial identities of R This section describes the monomial Type 1-identities for elementary gradingin R when the neutral component and diagonal coincide. The number s denotes | Supp G ( R ) | and λ denotes the number [ s + 1][( s + 1)( P si =1 ( s − i ) + 1]. Definition 5.1.
Let m = x i · · · x i q . Let k, l be two positive integers such that ≤ k ≤ l ≤ q . We define the monomial m [ k,l ] obtained from m by deleting the k − first variables and the q − l last variables. Definition 5.2.
Let S denote the set of all sequences with elements in Supp G ( R ) whose length is less than s + 1 . Let A = { ( g , . . . , g m ) ∈ S| g . . . . .g m = e } . Remark 5.3.
Notice that |S| = P si =1 s i . Definition 5.4.
A monomial m = x i · · · x i l is called a Type 2-monomial whenthere exist a ∈ N − { } , p , p ∈ b l such that ≤ p < p + a < p < p + a ≤ l and: : α ( x i p · · · x i p a ) = α ( x i p a +1 · · · x i p − ) = α ( x i p · · · x i p a ) = e ; : h ( x i p · · · x i p a ) = h ( x i p · · · x i p a ) . Definition 5.5.
Let m = x i · · · x i l be a monomial. It is called a Type 3-monomialwhen it does not have a proper subword of G -degree e . Otherwise, it is called a Type4-monomial. Corollary 5.6.
Let m = x i · · · x i l be a Type 1-monomial without variables of G -degree e . If l > s , then m is a Type 4- monomial.Proof. Let β ( t ) = α ( m [1 ,t ] ) be a function with domain b l and codomain Supp G ( R ).For the hypothesis, l > s . Consequently, according to the Pigeonhole Principle,there exists 1 ≤ t < t ≤ l such that β ( t ) = β ( t ). Note that t + 1 < t because m does not have variable of G -degree e . So, m [ t +1 ,t ] satisfies the thesis statementof the corollary. (cid:3) ATRICES OVER AN INFINITE INTEGRAL DOMAIN 7
Lemma 5.7.
Let S be a multiset formed by elements of \ ( P si =1 ( s − i ) . If | S | ≥ ( s + 1)( P si =1 ( s − i )+ 1 , then there exists i ∈ \ ( P si =1 ( s − i ) such that this positiveinteger repeats, at least, s + 2 times in S .Proof. It is the immediate consequence of the Pigeonhole Principle. (cid:3)
Lemma 5.8.
Let m = x · · · x r be a multilinear Type 1-monomial without variablesof G -degree e . If the ordinary degree of m is greater than or equal to λ , then it isa Type 2-monomial.Proof. Let m be a multilinear Type 1-monomial of K h X i without variables of G -degree e whose ordinary degree is greater than or equal to ( s + 1) ( P si =1 ( s − i ) +( s + 1) and a denotes the number ( s + 1)( P si =1 ( s − i ) + 1.Let: m = m [1 ,s +1] , m = m [ s +2 , s +1)] , · · · , m a = m [( a − s +1)+1 ,a. ( s +1)] .By Corollary 5.6, for each m i , there is a proper subword of G -degree e and ordinarydegree less than or equal to s .Let γ : { m , · · · , m a } → A be a relation that assigns: ( g , · · · , g n ) ∈ γ ( m i ) if,and only if, there exists a subword of m i of ordinary degree n , m i, , such that h ( m i, ) = ( g , · · · , g n ). By Lemma 5.7, there exist subwords m i , , · · · , m i s +2 , ∈{ γ ( m ) , · · · , γ ( m a ) } of m i , · · · , m i s +2 ( i < · · · < i s +2 ) such that h ( m i , ) = · · · = h ( m i s +2 , ). By Pigeonhole Principle, there exist k, k + l ∈ { , · · · , s + 2 } such thatthe subword of m ( m i k ,i k + l , ), between m i k , and m i k + l , , with G -degree e .Therefore, m i k , m i k ,i k + l , m i k + l , is a Type 2- monomial. So, m is Type 2-monomial as well. (cid:3) Definition 5.9.
We denote by U the T G -ideal generated by the following identitiesof R : • x x − x x when α ( x ) = α ( x ) = e (1) ; • x x x − x x x when α ( x ) = α ( x ) = ( α ( x )) − = e (2) ; • x when R α ( x ) = { } (3) ; • The multilinear Type 1- monomial identities whose ordinary degrees are lessthan or equal to λ (4) . Recall that if m is a monomial, then m ∈ T G ( R ) if and only if m ∈ T G ( M n ( K ))(Remark 3.6). In the next lemma we follow an idea of ([4], Proposition 4.2). Lemma 5.10.
Let m = x · · · x q be ( q > λ ) a multilinear monomial. If m is aType 1- monomial identity for R , then m follows from (4) .Proof. Let m = x · · · x q be a multilinear Type 1-monomial identity for R where q ≥ λ + 1. We may suppose without any loss of generality that α ( x i ) = e for all i ∈ b q (Lemma 4.5). The proof is made by induction on q . Suppose that q = λ + 1.According to Lemma 5.8, m is a Type 2-monomial.Here, we use the same notation as in Lemma 5.8. If x p · · · x p + a is a gradedmonomial identity for R , then m is a consequence of the monomial identities of type(4). Thus, we may suppose that x p · · · x p + a / ∈ T G ( R ). Let b m = m [1 ,p − m [ p + a +1 ,q ] .If b m is a monomial identity for R , then m is a consequence of b m that is a Type1-monomial. Suppose for contradiction that b m is not a polynomial identity for R .We may suppose without loss of generality that p + a + 1 < p and q ≥ p + a + LU´IS FELIPE GONC¸ ALVES FONSECA
1. Therefore, there exist the matrix units e l k ∈ M n ( K ) α ( x ) , · · · , e l p − k p − ∈ M n ( K ) α ( x p − ) , e l p a +1 k p a +1 ∈ M n ( K ) α ( x p a +1 ) , · · · , e l q k q ∈ M n ( K ) α ( x q ) suchthat ( e l k · · · e l p − k p − ) . ( e l p a +1 k p a +1 . · · · .e l q k q ) = 0.Note that k p − = l p + a +1 = k p − = l p = k p + a . In this form, consider thefollowing evaluation in m : x i = e l i k i ∀ i ∈ b q − { p , · · · , p + a } , x l p j k p j = e l p j k p j ∀ j ∈ { , , · · · , a } .Thus ( e l k · · · e l p − k p − ) . ( e l p k p · · · e l p a k p a ) . ( e l p a +1 k p a +1 · · · e l q k q ) = 0.This is a contradiction, because m ∈ T G ( R ). By induction on q , the resultfollows. (cid:3) Lemma 5.11. If m is a Type 1-monomial identity of R , then m follows from (4) .Proof. It follows from Remark 3.5 and Lemma 5.10. (cid:3) The main result
The next lemma follows an idea of ([2], Lemma 5), ([3], Lemma 5), ([25], Lemma4) and ([23], Lemma 4.6).
Lemma 6.1.
Let m ( x , · · · , x q ) and n ( x , · · · , x q ) be two monomials such that thematrices n ( A , · · · , A q ) and m ( A , · · · , A q ) have in same position the same non-zero entry. Then: m ( x , x , · · · , x q ) ≡ n ( x , x , · · · , x q ) mod J .Proof. Let m = x i · · · x i r . According to Lemma 3.9, m − n is a multi-homogeneouspolynomial. Let m and n be two multilinear monomials (with the same variables)such that h ( m ) = h ( m ) and h ( n ) = h ( n ). Note that, it is enough to prove: m ≡ n mod J .We may suppose that m = x · · · x r . Therefore, there exists σ ∈ S r such that n = x σ (1) · · · x σ ( r ) . By this hypothesis, there is a position ( i, j ) ∈ b n × b n such that e i m ( A , · · · , A q ) e j = e i n ( A , · · · , A q ) e j = 0.Suppose that the entry of m ( A , · · · , A q ), in the position ( i, j ) is: y q α ( x ) , · · · y q r α ( x r ) ,r where q , · · · , q r ∈ b n and q = i . Therefore: e q s q α ( x · · · e q r s qrα ( xr ) = e q σ (1) s qσ (1) α ( xσ (1)) · · · e q σ ( r ) s qσ ( r ) α ( xσ ( r )) = e ij .In this form, there exist matrix units e i j ∈ M n ( K ) α ( x ) , · · · , e i r j r ∈ M n ( K ) α ( x r ) having the following property: e i j · · · e i r j r = e i σ (1) j σ (1) · · · e i σ ( r ) j σ ( r ) = 0.So, i = i σ (1) , j r = j σ ( r ) and α ( m ) = α ( n ) = g − i g j .In the following steps, we will be use an induction on r . If r = 1, the proof isobvious. : Step 1: Suppose that σ (1) = 1. In this case, the monomials m and n start with the same variable. Let m and n be two monomials obtainedfrom m and n respectively by deleting the first variable. By Lemma3.4, m ( A , · · · , A q ) and n ( A , · · · , A q ) have, in the same position, thesame non-zero entry. Hence, by induction hypothesis, m ≡ n modulo J .Consequently, m ≡ n modulo J as required. ATRICES OVER AN INFINITE INTEGRAL DOMAIN 9 : Step 2: Suppose that σ (1) >
1. Let t be the least positive integer such that σ − ( t + 1) < σ − (1) ≤ σ − ( t ). We define: k := σ − ( t + 1), k := σ − (1)and k = σ − ( t ). Note that: σ ( k ) = t + 1, σ ( k ) = 1, σ ( k ) = t , i σ ( c +1) = j σ ( c ) and i c +1 = j c for all c ∈ [ r −
1. It is clear that: n = x σ (1) · · · x σ ( r ) = n [1 ,k − n [ k ,k − n [ k ,k ]1 n [ k +1 ,r ]1 .Likewise: α ( n [1 ,k − ) = g − i σ (1) g j σ ( k − = g − i g i σ ( k = g − i g i t +1 ; α ( n [ k ,k − ) = g − i σ ( k g j σ ( k − = g − i t +1 g i σ ( k = g − i t +1 g i ; α ( n [ k ,k ]1 ) = g − i σ ( k g j σ ( k = g − i g j t = g − i g i t +1 .Thus, by the identities (1) and (2), it is possible to conclude that: n ≡ n [ k ,k ]1 n [ k ,k − n [1 ,k − n [ k +1 ,r ]1 mod J .Conclusion: n is a congruent monomial that starts with the same variableof m . Repeating the arguments of the first case, we conclude that m ≡ n modulo J . (cid:3) Lemma 6.2.
Let G be a finite group of order n . Then R does not satisfy a G -gradedmonomial identity.Proof. According to Remarks 3.5 and 3.6, it is sufficient to prove that M n ( K ) doesnot satisfy a multilinear monomial identity x . . . . .x l .It is clear that Supp G ( M n ( K )) = G . So, it is enough to prove that M n ( K )does not satisfy a Type-1 multilinear monomial identity. If l = 1, the proof isobvious. The proof is made by induction on l . According to the hypothesis ofinduction, there exist matrix units e i j ∈ ( M n ( K )) α ( x ) , · · · , e i l j l ∈ ( M n ( K )) α ( x l ) such that e i j · · · e i l j l = e i j l . Notice that there exists g k ∈ { g , . . . , g n } such that g − k g i = α ( x ). So, e ki · · · e i l j l = 0. The proof of Lemma 6.2 is complete. (cid:3) Lemma 6.3. If R does not satisfy a monomial identity, then T G ( R ) = J .Proof. Suppose for contradiction there exists f ( x , · · · , x t ) = P li =1 λ i m i ∈ T G ( R ) − J , where for all i ∈ b l : λ i ∈ K − { } , m i is a monomial. According to Lemma 3.7,we may suppose that f is a multi-homogeneous polynomial. Moreover, it can besupposed that l is the least positive integer with the following set: B = { q ∈ N | P qi =1 γ i n i ( x , · · · , x t ) ∈ T G ( R ) − J ; γ i ∈ K − { } for all i ∈ b q } .It is clear that m i ( A , · · · , A t ) = 0, for i = 1 , · · · , l , because R does not satisfya monomial identity. Furthermore, there exists k ∈ { , · · · , l } such that: m ( A , · · · , A t ) and m k ( A , · · · , A t )have, in the same position the same non-zero entry. Thus by Lemma 6.1, it followsthat m ≡ m k modulo J . Consequently, h = f + λ k ( m − m k ) ∈ T G ( R ) − J . Thecontraction, in addition the number of non-zero summands in h is less than l . (cid:3) Corollary 6.4.
Let K be an infinite integral domain. The Z n -graded polynomialidentities of M n ( K ) follow from: • x x − x x when α ( x ) = α ( x ) = 0 (1) ; • x x x − x x x when α ( x ) = α ( x ) = − ( α ( x )) = 0 (2) . Proof.
It follows from Lemmas 6.2 and 6.3. (cid:3)
Following word for word the work of Vasilovsky in [26] (see Lemma 1, Lemma 3and Corollary 4), we have the following lemma:
Lemma 6.5.
Let K be an infinite integral domain and m = x · · · x l be a multilinearType 1-monomial. Then m / ∈ T Z ( M n ( K )) . Corollary 6.6.
Let K be an infinite integral domain and m be a Type 1-monomial.Then m / ∈ T Z ( M n ( K )) . Following word for word the proof of Lemma 6.3, we have the following lemma:
Lemma 6.7. If R does not satisfy a Type 1-monomial identity, then T G ( R ) = J . Corollary 6.8.
Let K be an infinite integral domain. The Z -graded polynomialidentities of M n ( K ) follow from: • x x − x x when α ( x ) = α ( x ) = 0 (1) ; • x x x − x x x when α ( x ) = α ( x ) = − ( α ( x )) = 0 (2) ; • x when | α ( x ) | ≥ n (3) .Proof. It follows from Corollary 6.6 and Lemma 6.7. (cid:3)
Now, we present the main result of this paper.
Theorem 6.9.
Let G be an arbitrary group. Then T G ( R ) = U . If K is an infiniteintegral domain, then T G ( M n ( K )) = U .Proof. According to Lemmas 4.6 and 5.11, if m is a monomial identity of R , then m is consequence of (3) or (4). Thus, it is sufficient to imitate the proof of Lemma6.3 and replace J with U , early in the proof. (cid:3) Matrix-units graded identities of M n ( K ) over an infinite integraldomain The algebra M n ( K ) has a natural grading by M U n , the semigroup of matrixunits of class n . Definition 7.1.
Let
M U n = { ( i, j ) ∈ b n × b n } ∪ { } denote the semigroup of matrixunits of class n whose multiplication is defined as follows: : . ( i, j ) = ( i, j ) . ; : ( i, j )( k, l ) = ( i, l ) when j = k ; : ( i, j )( k, l ) = 0 when j = k . Let x and x ij , y ij , z ij denote the free variables whose M U n -degree are 0 and( i, j ), respectively. The following theorem addresses this issue: Theorem 7.2. ( [4] , Theorem 4.9) Let K be a field of characteristic zero. Then,the M U n -graded identities of M n ( K ) follow from: x ii y ii − y ii x ii when i ∈ b n (5) ; x ij y ji z ij − z ij y ji x ij when ≤ i, j ≤ n i = j (6) ; x (7) . ATRICES OVER AN INFINITE INTEGRAL DOMAIN 11
Here, we extend this result for infinite integral domains. Let J MU n be the T MU n -ideal generated by (5), (6) and (7).Let h ∈ M U n − { } . Let W h = { w (1) h , · · · , w ( n ) h , · · · } denote a countable setof commuting variables. Let W = S h ∈ MU n −{ } W h and let Ω = K [ W ] be thepolynomial ring in commuting variables of the set W . Definition 7.3.
A generic matrix of M n (Ω ) of M U n -degree ( i, j ) is a homoge-neous element of the following type: A ( k )( i,j ) := w ( k )( i,j ) e ij where ≤ i, j ≤ n and k ∈ { , , · · · } .The M U n -graded subalgebra generated by the generic matrices of M n (Ω ) is calledthe algebra of generic matrices which we denote by R . Notice that if m is a M U n -graded monomial identity of R , then m follows from(7). The main steps of the proof of Lemmas 6.1 and 6.3 hold also for this gradingand we obtain the following result. Theorem 7.4.
The
M U n -graded identities of R follow from: • x ii y jj − y jj x ii when i ∈ b n (5); • x ij y ji z ij − z ij y ji x ij when ≤ i, j ≤ n, i = j (6); • x (7) . If K is an infinite integral domain, then T MU n ( M n ( K )) = J MU n . In the rest of this paper we will only consider matrices over an infinite integraldomain. 8. Z p -graded central polynomial of M p ( K )Now, we describe the Z p -graded central polynomials of M p ( K ) when p is a primenumber.Let π : Z → Z n denote the canonical projection and let j ∈ Z n . The followingconvention will be described in this section: y jh,i := y kh,i when k = π − ( j ) ∩ b n . Definition 8.1. ( [8] , Preliminaries) A sequence ( γ , · · · , γ n ) of elements of Z n iscalled a complete sequence when the following conditions are satisfied: : γ + · · · + γ n = 0 ; : { γ , γ + γ , · · · , γ + · · · + γ n } = Z n . The next lemma is the immediate consequence of the complete sequence defini-tion.
Lemma 8.2.
A sequence ( γ , · · · , γ n ) of elements of Z n is a complete sequence of Z n if and only if there exist matrix units e i j ∈ M n ( K ) γ , · · · , e i n j n ∈ M n ( K ) γ n such that i l +1 = j l for all ( l + 1) ∈ b n . Moreover, b n = { i , · · · , i n } and i = j n . Following word for word the proof of Brand˜ao J´unior in [8] (see Lemma 1 andProposition 1), we have the following lemma:
Lemma 8.3.
The Z n -graded multilinear polynomial: P σ ∈ H n x σ (1) · · · x σ ( n ) ,where ( α ( x ) , · · · , α ( x n )) is a complete sequence of Z n , is a Z n -graded centralpolynomial of M n ( K ) . Furthermore, it is not a Z n -graded polynomial identity of M n ( K ) . Lemma 8.4.
Let m = x i · · · x i q be a monomial such that α ( m ) = 0 and α ( x i j ) = h i j for all j ∈ b q . Let y s ki h i ,i · · · y s kiq h iq ,i q be an entry of A i . · · · .A i q . If there existsa subsequence ( s ki v , · · · , s ki vn ) of ( s ki , · · · , s ki q ) such that { s ki v , · · · , s ki vn } = b n and s ki v = s ki , then there exist monomials m , · · · , m n such that ( α ( m ) , · · · , α ( m n )) is a complete sequence of Z n and m = m m · · · m n .Proof. If n = 1, the proof is obvious. From now on, n >
1. First, we assume that m is a multilinear monomial.In fact, there are matrix units e j l ∈ M n ( K ) α ( x ) , · · · , e j q l q ∈ M n ( K ) α ( x q ) suchthat e j l · · · e j q l q = e j l q , where j t = s ki t for all t ∈ b q . Likewise, j = l q because m ∈ K h X i . We may suppose without any loss of generality that j t = s ki vt for t = 2 , · · · , n .Let: m i = x i , i = 1 , · · · , n − m n = x n · · · x q .Notice that e j l e j l · · · e j n l q = e j l q . So { j , · · · , j n } = b n , l t = j t +1 for all t ∈ [ n − j = l q . Consequently, by Lemma 8.2, it follows that m , · · · , m n satisfy the thesis of this lemma.Now, we assume that m is an arbitrary monomial. We would choose a multilinearmonomial m such that h ( m ) = h ( m ). There exist monomials m , · · · , m n suchthat ( α ( m ) , · · · , α ( m n )) is a complete sequence of Z n and m = m m · · · m n .Thus, there must exist monomials m , · · · , m n such that m = m . · · · .m n and h ( m ) = h ( m ) , · · · , h ( m n ) = h ( m n ). The proof is complete. (cid:3) The proof of the following lemma is left as an exercise.
Lemma 8.5.
Let A = { a , · · · , a l } Z p be a set. Then: { a + i, . . . , a l + i } 6 = { a + j, . . . , a l + j } for any i, j ∈ Z p distinct. The next lemma is well known.
Lemma 8.6.
Let z , z ∈ K h X i . Then the monomials z and z z are Z -gradedcentral monomials of M ( K ) . Lemma 8.7.
Let p > . Let x , x be variables such that α ( x ) = α ( x ) = 0 . Then ( x x ) p ≡ x p x p mod T Z p ( M p ( K )) .Proof. Let A ∈ R α ( x ) and A ∈ R α ( x ) be two generic matrices. By Lemma 3.2,it is obvious that all positions in the diagonal of ( A A ) p (respectively A p A p )have non-zero entries. According to Lemma 6.1, it is sufficient to prove that e ( A A ) p = e ( A p A p ).In fact, e ( A A ) p = e ( P pi =1 y iα ( x ) , y i + α ( x ) α ( x ) , e iπ − ( i +2 α ( x )) ∩ b p ) p =( Q pi =1 ( y iα ( x ) , y i + α ( x ) α ( x ) , ) e ) = ( Q pi =1 y i + α ( x ) α ( x ) , )( Q pi =1 y iα ( x ) , ) e =( Q pi =1 y iα ( x ) , )( Q pi =1 y iα ( x ) , ) e = ( A ) p e ( A ) p e = A p A p e .So, ( x x ) p ≡ x p x p mod T Z p ( M p ( K )) as required. (cid:3) Lemma 8.8. ( [8] , Lemma 8) Let p > and let l ∈ [ p − . Let m = x p · · · x pl be a Z p -graded monomial such that α ( x i ) = α ( x j ) for i = j . Then m ∈ C Z p ( M p ( K )) . ATRICES OVER AN INFINITE INTEGRAL DOMAIN 13
Proof.
First, we assume that l = 1.According to Lemma 3.2( A α ( x )1 ) p = P pi =1 y iα ( x ) , y i + α ( x ) α ( x ) , . . . y i +( p − α ( x ) α ( x ) , e ii .So y iα ( x ) , y i + α ( x ) α ( x ) , . . . y i +( p − α ( x ) α ( x ) , = y jα ( x ) , y j + α ( x ) α ( x ) , . . . y j +( p − α ( x ) α ( x ) , for all i, j ∈ b p because h α ( x ) i = Z p . Consequently x p ∈ C Z p ( M p ( K )). Bearing in mind that C Z p ( M p ( K )) is a subalgebra, the result follows. (cid:3) Definition 8.9.
Let V denote the T Z p -graded space generated by the monomialsreported in the hypothesis of Lemma 8.6 or the monomials that satisfy the hypothesisof Lemma 8.8. The proofs of the two following lemmas (Lemmas 8.10 and 8.11) are left as anexercise.
Lemma 8.10.
Let z , · · · , z n ∈ K h X i . Let m = z .k · · · z .k n n , where k , · · · , k n ∈ N −{ } . Then there exists m ∈ h V i T Z such that m − m ≡ modulo T Z ( M ( K )) . Lemma 8.11.
Let p > and let x , · · · , x n ∈ K h X i i , i = 0 . Let m = x p.k · · · x p.k n n be a monomial, where k , · · · , k n ∈ N − { } . Then there exists m ∈ h V i T Z p suchthat m − m ≡ modulo T Z p ( M p ( K )) . Lemma 8.12.
Let m = x i · · · x i r ∈ C Z p ( M p ( K )) . Then there exists m ∈ h V i T Z p such that m − m ≡ modulo T Z p ( M p ( K )) .Proof. Notice that m / ∈ T Z p ( M p ( K )). Moreover, at least one variable of m has Z p -degree different than 0.By hypothesis, m is a Z p -graded central monomial of M p ( K ). Consequently,using the Lemma 3.2, it follows that: y iα ( x i ) ,i y i + α ( x i ) α ( x i ) ,i · · · y i + α ( x i + ··· x ir − ) α ( x ir ) ,i r = y jα ( x i ) ,i y j + α ( x i ) α ( x i ) ,i · · · y j + α ( x i + ··· x ir − ) α ( x ir ) ,i r for any i, j ∈ b p .Let x l , · · · , x l q be all the different variables of the monomial m . k i denotes theordinary degree of m with respect to variable x l i . Note that, for each i ∈ b q , k i is amultiple of p .Case 1: α ( x l i ) = 0 for all i ∈ b q . Let x k l · · · x k q l q be a monomial and let A l ∈ R α ( x l ) , · · · , A l q ∈ R α ( x lq ) be generic matrices. Evidently, the matrices A k l · · · A k q l q and m ( A l , · · · , A l q ) have in position (1 , m ≡ x k l · · · x k q l q mod J . Applying Lemma 8.10 or Lemma 8.11, weare done.Case 2: there exists i ∈ b q such that α ( x l i ) = 0. Suppose that all variables of G -degree 0 are { x l , · · · , x l s } . Choose m = ( x l s +1 x k l l · · · x k s l s ) p x k s +1 − pl s +1 x k s +2 l s +2 · · · x k lq l q .Evidently, m ( A l , · · · , A l q ) and m ( A i , · · · , A i r ) have in position (1 ,
1) the samenon-zero entry. Applying the ideas of previous case, the result follows. (cid:3)
Lemma 8.13.
Let m = x i · · · x i q ∈ K h X i − ( C Z p ( M p ( K ))) ∩ ( K h X i ) . All entriesin the diagonal of A i . · · · .A i q are non-zero and pairwise distinct. Proof. (Sketches) According to Lemma 3.2, all entries in the diagonal are non-zero.If p = 2, the analysis is obvious.Henceforth, suppose that p >
2. By hypothesis, x i · · · x i q ∈ K h X i − C Z p ( M p ( K )) ∩ K h X i . Thus, the following condition is satisfied: there exist j < · · · < j l ∈ b q ,where x i j = · · · = x i jl and x i t = x i j for all t ∈ b q − { j , . . . , j l } such that: y k + α ( x i )+ ··· + α ( x ij − ) α ( x ij ) ,i j · · · y k + α ( x i )+ ··· + α ( x ijl − ) α ( x ijl ) ,i jl = y k + α ( x i )+ ··· + α ( x ij − ) α ( x ij ) ,i j · · · y k + α ( x i )+ ··· + α ( x ijl − ) α ( x ijl ) ,i jl . By Lemma 8.5 and some calculations, we may conclude that: y q + α ( x i )+ ··· + α ( x ij − ) α ( x i ) ,i · · · y q + α ( x i )+ ··· + α ( x ijl − ) α ( x ijl ) ,i jl = y q + α ( x i )+ ··· + α ( x ij − ) α ( x i ) ,i · · · y q + α ( x i )+ ··· + α ( x ijl − ) α ( x ijl ) ,i jl for all q = q ∈ Z p .The proof is complete. (cid:3) In what follows, we use substantially the proof of Theorem 6 used by Brand˜aoJ´unior [8].
Theorem 8.14.
The Z p -graded central polynomials of M p ( K ) follow from: z ( x x − x x ) z when α ( x ) = α ( x ) = 0 (8); z ( x x x − x x x ) z when α ( x ) = − α ( x ) = α ( x ) = 0 (9); The monomials cited in the definition 8.9 (10) ; P σ ∈ H p x σ (1) · · · x σ ( p ) , where ( α ( x ) , · · · , α ( x p )) is a complete sequence of Z p (11) .The monomials z , z ∈ S g ∈ Z p X g .Proof. Let W be the T Z p -space generated by (8) , (9) , (10) and (11). We prove that C Z p ( M p ( K )) ⊂ W . Let f ( x , · · · , x q ) = P li =1 λ i m i ∈ C Z p ( M p ( K )) − T Z p ( M p ( K )).By Lemma 3.8 and Corollary 3.11, we may assume that f is a multi-homogeneouspolynomial. We may suppose that α ( m ) = · · · = α ( m l ) = 0 , m i − m j is not anelement of T Z p ( M p ( K )) and each m i / ∈ C Z p ( M p ( K )) (Lemma 8.12).Let A ∈ R α ( x ) , · · · , A q ∈ R α ( x q ) be generic matrices. So f ( A , · · · , A q ) = diag ( F , · · · , F p ), where F = · · · = F p = 0. By Lemma 8.13, for each j ∈ b p ,all positions in the diagonal of the matrix m j ( A , · · · , A q ) have non-zero entries.Furthermore, these entries are pairwise distinct.Reordering the indices, if necessary, there exist 1 ≤ i < · · · < i p ≤ l such that λ i = · · · = λ i p = 0 and e m i ( A , · · · , A q ) e = e l m i l ( A , · · · , A q ) e l for all l ∈ b p − { } . Assume that m i = x j · · · x j s and the entry in position (1 ,
1) of m i ( A , · · · , A q ) is y a α ( x j ) ,j · · · y a s α ( x js ) ,j s . Notice that the multi-set { a , · · · , a s } contains b p .According to Lemma 8.4, there are monomials r , · · · , r p such that m i = r · · · r p where ( α ( r ) , · · · , α ( r p )) is a complete sequence of Z p . For each j ∈ b p , there is aunique permutation σ ∈ H p such that the matrices: m i j ( A , · · · , A q ) and r σ (1) · · · r σ ( p ) ( A , · · · , A q )have, in the position (1 , m i j ≡ r σ (1) · · · r σ ( p ) mod T Z p ( M p ( K )). ATRICES OVER AN INFINITE INTEGRAL DOMAIN 15
According to Lemma 8.3, it is clear that: g ( x , · · · , x r ) = λ i ( P σ ∈ H p r σ (1) · · · r σ ( p ) ) ∈ W .Then, f − g ≡ f − λ i ( m i + m i + · · · + m i p ) modulo T Z p ( M p ( K )). If l − p = 0,it follows that f ∈ W . If l − p ≥ p or 1 ≤ l − p ≤ p −
1, the same argument can berepeated. From an inductive argument on l , the result follows. (cid:3) Z -graded central polynomials of M n ( K )In this section, we use the same technique as in previous section to detail a scriptof the proof. The first Lemma is similar to Lemma 8.13. Lemma 9.1.
Let x i . · · · .x i m ∈ K h X i .If A α ( x i ) i . · · · .A α ( x im ) i m is a non-zero matrix, then all non-zero entries of thatmatrix are pairwise distinct.Proof. (Sketches) If only one position in A i . · · · .A i m has a non-zero entry, the proofis obvious. Suppose that there exist, at least, two positions ( k , k ) , ( k , k ) ∈ b n × b n such that: e k ( A i · · · A i m ) e k , e k ( A i · · · A i m ) e k = 0.Our aim is to prove that: e k ( A i · · · A i m ) e k = e k ( A i · · · A i m ) e k .Suppose by contradiction that: e k ( A i · · · A i m ) e k = e k ( A i · · · A i m ) e k .By contradiction, the result follows. (cid:3) The main steps of the proof of the Lemmas 8.3, Lemma 8.4 and Theorem 8.14hold also for this grading and we obtain the following result.
Theorem 9.2.
Let K be an infinite integral domain. Then the Z -graded centralpolynomials of M n ( K ) follow from: z ( x x − x x ) z when α ( x ) = α ( x ) = 0 (12) ; z ( x x x − x x x ) z when α ( x ) = − α ( x ) = α ( x ) = 0 (13) ; z x z when | α ( x ) | ≥ n (14) ; P σ ∈ H n x σ (1) · · · x σ ( n ) , where ( α ( x ) , · · · , α ( x n )) is a complete sequence of Z n and | α ( x i ) | < n (15) .The monomials z , z ∈ S i ∈ Z X i . Acknowledgments
My sincerely thanks go to Alexei N. Krasilnikov, for proposing the problem, andmy entire doctorate board (Alexei Krasilnikov, Plamen Koshlukov, Viviane RibeiroTomaz da Silva, Victor Petrogradsky and Jos´e Antˆonio Oliveira Freitas) for theiruseful advice, comments, remarks and suggestions.The author would like to thank the reviewers of Rendiconti del Circolo Matem-atico di Palermo for their comments that help improve the manuscript.
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Departamento de Matem´atica, Universidade Federal de Vic¸osa - Campus Florestal,35690-000 - Florestal
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