Idempotent factorization of matrices over a Prüfer domain of rational functions
aa r X i v : . [ m a t h . A C ] J u l IDEMPOTENT FACTORIZATION OF MATRICES OVER APR ¨UFER DOMAIN OF RATIONAL FUNCTIONS
LAURA COSSU
Abstract.
We consider the smallest subring D of R ( X ) containing every el-ement of the form 1 / (1 + x ), with x ∈ R ( X ). D is a Pr¨ufer domain calledthe minimal Dress ring of R ( X ). In this paper, addressing a general openproblem for Pr¨ufer non B´ezout domains, we investigate whether 2 × D can be decomposed as products of idempotent matrices. Weshow some conditions that guarantee the idempotent factorization and othersthat forbid easy decompositions in M ( D ). Introduction
In 1965 Andreas Dress [7] introduced a family of Pr¨ufer domains constructed assubrings D K of a field K containing every element of the form 1 / (1+ x ), for x ∈ K .Given a field K not containing square roots of −
1, the subring of K generated by { (1 + x ) − : x ∈ K } is said to be the minimal Pr¨ufer-Dress ring (or simply the minimal Dress ring ) of K . We refer to [7] and [4] for more details on these domains.In the paper [4], the authors investigated minimal Dress rings of special classes offields: Henselian fields, ordered fields and formally real fields (e.g., R ( A ), with A a set of indeterminates). They focused in particular on the minimal Dress ring D of the field of real rational functions R ( X ), characterizing its elements [4, Prop.2.1] and ideals [4, Prop. 2.4] and proving that D is a Dedekind domain (i.e., aNoetherian Pr¨ufer domain) that is not a principal ideal domain [4, Th. 2.3]. Theyalso identified a family of 2 × D that can be written asa product of idempotent factors [4, Th. 3.3]. The study of the factorization ofsingular square matrices over rings as product of idempotent matrices has raiseda remarkable interest both in the commutative and non-commutative setting sincethe middle of the 1960’s (see [8, 11]). We say that an integral domain R satisfiesthe property (ID ) if every 2 × R is a product of idempotentfactors. A natural and well motivated conjecture, proposed by Salce and Zanardoin [11] and then investigated in [3] and [5], asserts that every domain R satisfying(ID ) must be a B´ezout domain, namely, every finitely generated ideal of R mustbe principal. Note that the reverse implication is false: not every B´ezout domainverifies (ID ) (see [2, 6]). In [3] it is proved that if R satisfies (ID ), then everyfinitely generated ideal of R is invertible and so R is a Pr¨ufer domain. Therefore,it is not restrictive to study (ID ) within this class of domains and, in view ofthe above conjecture, we expect that for every Pr¨ufer non-B´ezout domain R there Mathematics Subject Classification.
Key words and phrases.
Factorization of matrices, idempotent matrices, minimal Dress rings,Pr¨ufer domains.The author is a member of the Gruppo Nazionale per le Strutture Algebriche, Geometriche e leloro Applicazioni (GNSAGA) of the Istituto Nazionale di Alta Matematica (INdAM). exists at least one singular matrix in M ( R ) that cannot be written as a productof idempotents.In this paper we develop the investigation started in [4] of the idempotent fac-torization of 2 × D of R ( X ). In Section 2 wefix the notation and recall some preliminary results and definitions. In Section 3 weconcentrate on the factorizations in M ( D ) and, in Theorems 3.3, 3.8 and 3.10, weidentify several conditions on a couple of elements p, q ∈ D under which the matrix (cid:18) p q (cid:19) factors into idempotents. Even if the problem whether D satisfies or not(ID ) remains open, our results contribute to narrow down the class of singulardimension 2 matrices over D that might not admit an idempotent factorization. InSection 4 we describe a family of matrices in M ( D ) that cannot be written as aproduct of two or three particular idempotent matrices.2. Preliminaries and notation
Let R be an integral domain. We will use the standard notations R × to denote itsmultiplicative group of units and M n ( R ) to denote the R -algebra of n × n matricesover R .A square matrix T over R is said to be idempotent if T = T . A direct com-putation shows that a singular nonzero matrix (cid:18) a bc d (cid:19) over an arbitrary integraldomain is idempotent if and only if d = 1 − a . It is also very easy to see that everymatrix similar to an idempotent matrix is idempotent, hence for a singular matrix S ∈ M n ( R ), the property of being a product of idempotent factors is preserved bysimilarity. This immediately leads to the following lemma. Lemma 2.1 (Lemma 3.1 of [4]) . Let R be a domain, p, q ∈ R . The matrix (cid:18) p q (cid:19) is a product of idempotent matrices if and only if such is (cid:18) q p (cid:19) . The next result will also be useful in the following.
Lemma 2.2.
Let p and q two nonzero elements of an integral domain R , and M = (cid:18) p q (cid:19) ∈ M ( R ) . If M = S · T , with S = (cid:18) p ′ q ′ z t (cid:19) a singular matrix and T = (cid:18) a bc − a (cid:19) an idempotent matrix over R , then S has the form S = (cid:18) p ′ q ′ (cid:19) . We omit the proof, since it is essentially contained in that of Lemma 3.1 in [6].Finally, we recall below two immediate factorizations in M ( R ):(1) (cid:18) p
00 0 (cid:19) = (cid:18) −
10 0 (cid:19) (cid:18) − p (cid:19) ; (cid:18) q (cid:19) = (cid:18) (cid:19) (cid:18) q (cid:19) . From now on D will denote the minimal Dress rings of the field of rationalfunctions R ( X ).Following the notation in [4], let Γ be the set of the polynomials in R [ X ] that haveno roots in R . Then Γ = { α Q i γ i } , where the γ i are monic degree-two polynomialsirreducible over R [ X ] and 0 = α is a real number. Set Γ + = { f ∈ R [ X ] : f ( r ) > , ∀ r ∈ R } , and, correspondingly, Γ − = {− f : f ∈ Γ + } . Then, by Proposition 2.1in [4], D = { f /γ : f ∈ R [ X ] , γ ∈ Γ + , deg f ≤ deg γ } , and D × = { γ /γ : γ ∈ Γ , γ ∈ Γ + , deg γ = deg γ } . As recalled in the introduction, we know from Theorem 2.3 and Proposition 2.4of [4] that D is a Dedekind domain which is not a principal ideal domain. As anexample, the ideal generated by 1 /γ and X/γ , with γ ∈ Γ + , is not principal.Given a polynomial f ∈ R [ X ] we will denote as l . c . ( f ) its leading coefficient.With a slight abuse of notation if p = f /γ , with f ∈ R [ X ] and γ ∈ Γ + , is anelement of D , we will refer to l . c . ( f ) as the leading coefficient of p .3. Idempotent factorizations in M ( D )In this section we focus on the investigation of property (ID ) over D . We givea considerable improvement to the findings in [4] by showing several families ofsingular matrices over D that enjoy a factorization into idempotents.Let us start recalling the main results of [4] on products of idempotent matricesover D . Lemma 3.1 (Lemma 3.2 of [4]) . Let x, y be two non-zero polynomials in R [ X ] with deg x = deg y .(a) If y ( u ) > (or y ( u ) < ) for every u root of x , then there exists β ∈ Γ suchthat δ = x + yβ ∈ Γ + , deg x − ≤ deg β ≤ deg x = deg δ/ .(b) If x ( z ) > (or x ( z ) < ) for every z root of y , then there exists η ∈ Γ suchthat δ = xη + y ∈ Γ + and deg y − ≤ deg η ≤ deg y = deg δ/ . Theorem 3.2 (Th. 3.3 of [4]) . Let p and q be two elements of D . Then the matrix (cid:18) p q (cid:19) is a product of idempotent matrices if one of the following holds:(i) deg p ≥ deg q and q ( u ) > (or q ( u ) < ) for every u root of p (ii) deg q ≥ deg p and p ( z ) > (or p ( z ) < ) for every z root of q . Two polynomials x, y ∈ R [ X ] are said to be weakly comaximal if gcd( x, y ) ∈ Γ,i.e., if x and y have no common roots. Let p and q be two elements of D . Then wecan always write p = x/γ and q = y/γ , with γ ∈ Γ + and x, y ∈ R [ X ]. We say that p and q are weakly comaximal if so are x and y . Theorem 3.3.
Let p and q be two weakly comaximal elements of D . If either p ≥ or q ≥ , then the matrix (cid:18) p q (cid:19) is a product of idempotent matrices.Proof. By Lemma 2.1, (cid:18) p q (cid:19) is a product of idempotents if and only if such is (cid:18) q p (cid:19) , therefore we can safely assume that p ≥ p ≥ deg q . We can assume without loss ofgenerality that deg p = deg q . In fact, since (cid:18) p q (cid:19) is similar to (cid:18) p p + q (cid:19) and p and p + q are still weakly comaximal, if deg p > deg q , it not restrictive to replace q LAURA COSSU with p + q . Thus, let deg p = deg q and set p = x/γ and q = y/γ , with x, y ∈ R [ X ], γ ∈ Γ + .As a further reduction, we may assume that deg p = deg q = 0. In fact, being p ≥
0, every root of p has even multiplicity and deg x is even. Moreover, if deg x < deg γ , taking any τ ∈ Γ + such that deg x = deg τ , (cid:18) x/γ y/γ (cid:19) = (cid:18) τ /γ
00 0 (cid:19) (cid:18) x/τ y/τ (cid:19) is a factorization in M ( D ) and, by (1), the matrix on the left of the above equalityis a product of idempotents if such is the second factor of the product on the right.Since for every z root of y , x ( z ) is always >
0, we have got in the position to applyLemma 3.1 (ii) to x and y . Therefore, there exists η ∈ Γ such that δ = xη + y ∈ Γ + where deg η = deg x and deg δ = 2deg η .Then, since deg x = deg y = deg γ = deg η , δ/γη ∈ D × and xη/δ , yη/δ , xy/δ , y /δ ∈ D . Moreover, the relation 1 − xη/δ = y /δ implies that T = (cid:18) xη/δ yη/δxy/δ y /δ (cid:19) is an idempotent matrix over D . Therefore (cid:18) x/γ y/γ (cid:19) = (cid:18) δ/γη
00 0 (cid:19) T , and using (1) we conclude that (cid:18) p q (cid:19) is a product of idempotent matrices over D .On the other hand, if deg q > deg p , it suffices to apply Theorem 3.2 (ii). (cid:3) Remark . The matrix (cid:18) p q (cid:19) ∈ M ( D ) is a product of idempotent matriceseven if p and q are two comaximal elements of D such that either p ≤ q ≤ Lemma 3.5.
Let x, y ∈ R [ X ] and ε ∈ R + be such that: • y has as unique root with multiplicity k ; • x (0) = 0 ; • y ( i ) > for ≤ i ≤ k − on (0 , ε ] ; • y ( k ) does not change sign on (0 , ε ] ; • x ( j ) for ≤ i ≤ k are either zero or do not change sign on (0 , ε ] .Then, there exists a real number r > such that rx + y has at most a unique rooton (0 , ε ] for every r ∈ (0 , r ] , and exactly one root if x < on (0 , ε ] .Proof. Note that y ( k ) (0) = 0, hence y ( k ) has nonzero max and min in [0 , ε ]. Since,by assumption, x ( k ) is either zero or does not change sign in (0 , ε ], an easy directcheck shows that, for all possible signs, there exists an r > rx ( k ) + y ( k ) is either strictly positive or negative in [0 , ε ], so has no roots, for every r ∈ (0 , r ].It follows that ∀ r ∈ (0 , r ], rx ( k − + y ( k − is either increasing or decreasing in theinterval, hence it has at most one root.Now, let us consider the ( k − rx + y .We distinguish three cases.(i) If x ( k − ≥ , ε ], then rx ( k − + y ( k − > r ∈ ( 0 , r ], therefore rx ( k − + y ( k − has at most a unique root in theinterval, being increasing. (ii) If x ( k − < , ε ] and y ( k − (0) = 0, then, by possibly choosing a smaller r , rx ( k − + y ( k − is either strictly positive or negative in [0 , ε ] for every r ∈ (0 , r ], and again we get that rx ( k − + y ( k − has at most a unique root.(iii) If x ( k − < , ε ] and y ( k − (0) = 0, by possibly choosing a smaller r , rx ( k − ( ε ) + y ( k − ( ε ) > r ∈ (0 , r ]. Since rx ( k − (0) + y ( k − (0) ≤
0, we have two possibilities: or rx ( k − + y ( k − > , ε ] for every r ∈ (0 , r ], or there exists x rk − ∈ (0 , ε ) such that rx ( k − ( x rk − )+ y ( k − ( x rk − ) = 0and this zero is unique since rx ( k − + y ( k − has at most one root. As aconsequence, in the first case rx ( k − + y ( k − is strictly increasing and admitsat most one root, in the second case it decreases on (0 , x rk − ) and increases on( x rk − , ε ].Let us now distinguish three more cases for the ( k − rx + y .(i) If x ( k − ≥ , ε ], then rx ( k − + y ( k − > r ∈ (0 , r ], therefore rx ( k − + y ( k − is increasing and it has at most aunique root in the interval.(ii) If x ( k − < , ε ] and y ( k − (0) = 0, then, by possibly choosing a smaller r , rx ( k − + y ( k − is either strictly positive or negative in [0 , ε ] for every r ∈ (0 , r ], and again we get that rx ( k − + y ( k − has at most a unique root.(iii) If x ( k − < , ε ] and y ( k − (0) = 0, by possibly choosing a smaller r rx ( k − ( ε )+ y ( k − ( ε ) > r ∈ (0 , r ]. Since rx ( k − (0)+ y ( k − (0) ≤ rx ( k − + y ( k − > , ε ] for every r ∈ (0 , r ]or there exists x rk − ∈ (0 , ε ), zero of rx ( k − + y ( k − for any r ∈ (0 , r ]. Bythe previous step rx ( k − + y ( k − is either increasing or has a unique criticalpoint on (0 , ε ]. In both this cases we cannot have other roots besides x rk − .As a consequence rx ( k − + y ( k − is either strictly increasing or it decreaseson (0 , x rk − ) and increases on ( x rk − , ε ].Iterating the procedure, after k steps we obtain that there exists a real number r > rx + y has at most a unique root on (0 , ε ] for every r ∈ (0 , r ], andexactly one root if x < , ε ]. (cid:3) Remark . In the hypothesis of the above Lemma, if k = 1, the proof becomesmuch easier. If x > , ε ], rx + y > r ∈ R . Let us assumehenceforth that x < , ε ]. There always exists a suitable r > rx ( ε ) + y ( ε ) > r ∈ (0 , r ]. Since y (0) = 0 and y ( ε ) >
0, it must be y ′ > , ε ]. If on the same interval x ′ ≥ rx ′ + y ′ > rx (0) + y (0) < rx + y has a unique root on (0 , ε ] for every r ∈ (0 , r ]. If x ′ < , ε ], by possiblychoosing a smaller r , rx ′ + y ′ is still strictly positive on (0 , ε ] for every r ∈ (0 , r ]and we conclude as before. Lemma 3.7.
Let x, y be polynomials in R [ X ] without common roots, such that deg x and deg y are odd, deg x > deg y , y has a unique root and there exist x , x ∈ R roots of x such that y ( x ) y ( x ) < . Then, there exists a suitable r ∈ R , such that rx + y has a unique root.Proof. It is not restrictive to assume lim X →±∞ xy = + ∞ . If the leading coefficientsof x and y are discordant the proof can be accordingly adapted by replacing r with − r .Let lim X →±∞ x, y = ±∞ . The case lim X →±∞ x, y = ∓∞ is analogous. Up to asuitable translation we can assume y (0) = 0, x < x >
0. Let k be the (odd) LAURA COSSU multiplicity of 0 as root of y . Let I = ( − ε, ε ) be a sufficiently small neighborhoodof 0 such that on it: y is strictly increasing and x <
0. The case x > − ε, ε , that on (0 , ε ], y ( i ) > ≤ i ≤ k − y ( k ) does not change sign and x ( j ) for 1 ≤ i ≤ k are either zero or do not change sign. By Lemma 3.5 there existsa real number r > r ∈ (0 , r ], rx + y has a unique root on[0 , ε ]. Let us observe that in [ − ε, rx + y < r .Now let M ∈ R + such that xy > X such that | X | > M . Clearly,for every r > rx + y has no roots for | X | > M . Let us consider the closedintervals I = [ − M, − ε ] and I = [ ε, M ]. Let M x = max I ∪ I | x | > m y = min I ∪ I | y | . Since y has no roots other than 0, clearly m y >
0. By choosing0 < r < m y /M x the polynomial rx + y has no zeroes on I ∪ I .We can conclude by choosing any 0 < r < min { r , m y /M x } . (cid:3) Theorem 3.8.
Let p and q be two elements of D . If deg p, deg q are odd andeither p or q has a unique root, then the matrix (cid:18) p q (cid:19) is a product of idempotentmatrices.Proof. By Lemma 2.1, we can safely assume that q has a unique root.We first consider the case p and q weakly comaximal.If deg q ≥ deg p , since q has a unique root and p and q have not common factors,the hypothesis of Th. 3.2 (ii) are satisfied.If deg p > deg q we distinguish two cases. If q does not change sign on the roots of p , we are done by Theorem 3.2 (i). Otherwise, by Lemma 3.7, it is always possibleto find a suitable r ∈ R such that rp + q has a unique root. Therefore, by Theorem3.2 (ii), the matrix (cid:18) p rp + q (cid:19) , similar to (cid:18) p q (cid:19) is a product of idempotentmatrices.Let us now consider the case p and q not weakly comaximal, i.e., if p = x/γ and q = y/γ , with γ ∈ Γ + and x, y ∈ R [ X ], x and y have a common root. Since q hasodd degree and a unique root z ∈ R , we have x = ( X − z ) h ¯ x and y = ( X − z ) k ¯ y with k, h positive integers, k odd, ¯ y ∈ Γ, ¯ x ∈ R [ X ] and gcd( X − z, ¯ x ) = 1. Letus choose any δ ∈ Γ + such that either deg δ = min { k, h } or deg δ = min { k, h } + 1according with the parity of min { k, h } . Since max { deg p, deg q } < (cid:18) p q (cid:19) = (cid:18) ( X − z ) min { k,h } /δ
00 0 (cid:19) (cid:18) ( X − z ) h − min { k,h } ¯ xδ/γ ( X − z ) k − min { k,h } ¯ yδ/γ (cid:19) is a fac-torization in M ( D ) and, by (1), (cid:18) p q (cid:19) is product of idempotent matrices if suchis S = (cid:18) ( X − z ) h − min { k,h } ¯ xδ/γ ( X − z ) k − min { k,h } ¯ yδ/γ (cid:19) . Let us remark that theelements of the first row of S , ( X − z ) h − min { k,h } ¯ xδ/γ and ( X − z ) k − min { k,h } ¯ yδ/γ ,are now weakly comaximal.If h ≥ k , S = (cid:18) ( X − z ) h − k ¯ xδ/γ ¯ yδ/γ (cid:19) and, since ¯ yδ ∈ Γ, we conclude byapplying Theorem 3.3. If k > h , S = (cid:18) ¯ xδ/γ ( X − z ) k − h ¯ yδ/γ (cid:19) . If h is even, since deg x = h + deg ¯ x is odd, then deg ¯ x is odd. Moreover also k − h is odd. It follows that ¯ xδ/γ and( X − z ) k − h ¯ yδ/γ are two weakly comaximal element of D with odd degree and, being¯ y and δ elements of Γ, ( X − z ) k − h ¯ yδ/γ has a unique root z ∈ R . Therefore, fromthe first part of the proof, we conclude that S is a product of idempotent matrices.If h is odd, being k − h even, ( X − z ) k − h ¯ yδ is always ≥ ≤ (cid:3) Lemma 3.9.
Let x, y be polynomials in R [ X ] without common roots, such that deg x is even, deg y is odd, deg x > deg y , y has a unique root y and there exist x , x ∈ R roots of x such that y ( x ) y ( x ) < . Then, there exists a suitable r ∈ R ,such that rx + y has exactly two distinct roots z , z ∈ R . Moreover, if the sign of x ( y ) and that of the leading coefficient of x are the same (resp. opposite), then x ( z ) x ( z ) > (resp. x ( z ) x ( z ) < ).Proof. It is not restrictive to assume lim X →±∞ x = + ∞ and lim X →±∞ y = ±∞ .As for Lemma 3.8, if the leading coefficients of x and y are discordant or bothnegative, the proof can be easily adapted.Up to a suitable translation we can assume y (0) = 0, x < x >
0. Let k be the (odd) multiplicity of 0 as root of y . Let I = ( − ε, ε ) be a sufficiently smallneighborhood of 0 such that on it x < , ε ] y ( i ) > ≤ i ≤ k − y ( k ) does not change sign and x ( j ) for 1 ≤ i ≤ k are either zero or do not change sign.The case x > − ε, r > r ∈ (0 , r ], rx + y has a unique root on [ 0 , ε ] for every r ∈ (0 , r ].Let us observe that in [ − ε, rx + y < r .Let M ∈ R + such that x, y > X > M . Clearly, for every r > rx + y has no roots for on ( M, N ∈ R + such that x > y < X ≤ − N and x ′ > y ′ < −∞ , − N ). Under these assumptions, there exists a real number r > r ∈ (0 , r ], rx ′ + y ′ < rx + y )( − N ) < rx + y has a unique root on ( −∞ , − N ] for every r ∈ (0 , r ].Let us consider the closed intervals I = [ − N, − ε ] and I = [ ε, M ]. Let M x = max I ∪ I | x | > m y = min I ∪ I | y | > y has no roots other than 0).By choosing 0 < r < m y /M x the polynomial rx + y has no zeroes on I ∪ I .We can conclude by choosing any 0 < r < min { r , r , m y /M x } .The last statement of the theorem follows immediately by construction. (cid:3) Theorem 3.10.
Let p and q be two weakly comaximal elements of D . If deg p iseven, deg q is odd and either p has a unique root or q has a unique root u such that p ( u ) has the same sign of the leading coefficient of p , then the matrix (cid:18) p q (cid:19) is aproduct of idempotent matrices.Proof. Let us start assuming that p has a unique root. Since p has even degree itis not restrictive to assume that p ≥
0, then we conclude by Theorem 3.3.Assume now that q has a unique root u and that p ( u ) has the same sign of theleading coefficient of p . We distinguish two cases. LAURA COSSU
If deg q > deg p , since q has a unique root and p and q have not common factors,we reach the thesis by applying Th. 3.2 (ii).If deg p > deg q we have two possibilities. If q does not change sign on the roots of p , we are done by Theorem 3.2 (i). Otherwise, by Lemma 3.9, it is always possibleto find a suitable r ∈ R such that rp + q has exactly two roots z , z such that x ( z ) x ( z ) >
0. Therefore, by Theorem 3.2 (ii), the matrix (cid:18) p rp + q (cid:19) , similarto (cid:18) p q (cid:19) is a product of idempotent matrices. (cid:3) Remark . Let p = x/γ and q = y/γ be elements of D such that max { deg p, deg q } =0. If p and q have a common factor M / ∈ Γ, whenever the degree of M is odd anddeg δ ≥ M , the decomposition (cid:18) p q (cid:19) = (cid:18) M/δ
00 0 (cid:19) (cid:18) xδ/M γ yδ/M γ (cid:19) is not a factorization in D since max { deg( xδ/M γ ) , deg( yδ/M γ ) } ≥
1. For this rea-son, we can not generalize Theorem 3.10 to the non-comaximal case as we have donein Theorem 3.8. However, under the additional hypothesis that max { deg p, deg q } <
0, the following corollary holds.
Corollary 3.12.
Let p = ( X − z ) k ¯ x/γ and q = ( X − z ) h ¯ y/γ , with k, h ∈ N + , ¯ x, ¯ y ∈ R [ X ] , γ ∈ Γ + , ¯ x ( z ) = 0 , ¯ y ( z ) = 0 be two elements of D such that max { deg p, deg q } < . If deg p is even, deg q is odd and either p has z as unique rootand sgn(¯ y ( z )) = sgn(l . c . (¯ y )) or q has z as unique root and sgn(¯ x ( z )) = sgn(l . c . (¯ x )) ,then the matrix (cid:18) p q (cid:19) is a product of idempotent matrices.Proof. We skip the details of the proof since it is analogous to the second part ofthe proof of Theorem 3.8. We reach the thesis by properly applying Theorems 3.3and 3.10 and using (1). (cid:3)
Remark . It is worth noting that the couples ( p, q ) ∈ D such that (cid:18) p q (cid:19) isa product of idempotent matrices, can generate both principal and non-principalideals of D . Thus, this characterization of the elements p and q is not related tothe idempotent factorization of (cid:18) p q (cid:19) . The same fact can be observed in [5] forthe factorization into idempotent factors of matrices of the form (cid:18) p q (cid:19) over realquadratic integer rings.4. A family of matrices in M ( D ) not admitting “easy” idempotentfactorizations As recalled in Section 2, the minimal Dress ring D of R ( X ) is a (Noetherian)Pr¨ufer non-B´ezout domain. In light of the conjecture mentioned in the introduction,we expect that there exists a counterexample to the fact that D verifies property(ID ). The results in the previous section contribute to contain the family of 2 × D to which such a counterexample might belong. In whatfollows we identify a class of matrices in M ( D ) which do not allow for simpleidempotent decompositions and that might not even admit one. Lemma 4.1.
Let p = x/γ and q = y/γ be two weakly comaximal elements of D .Let (cid:18) p q (cid:19) = (cid:18) p ′ q ′ (cid:19) (cid:18) a bc − a (cid:19) , with p ′ = x ′ /η, q ′ = y ′ /η, a = a ′ /δ, b = b ′ /δ, c = c ′ /δ ∈ D and a (1 − a ) = bc . Then there exist t, s ∈ R [ X ] , σ, ζ ∈ Γ + suchthat xt + ys = σ,x ′ t + y ′ s = ζ. Moreover, deg( xt ) , deg( xs ) , deg( yt ) , deg( ys ) ≤ deg σ .Proof. Since by assumption p = p ′ a + q ′ c , q = p ′ b + q ′ (1 − a ) and a (1 − a ) = bc , weget(2) p/q = a/b = c/ (1 − a ) . Set x = ¯ xθ , y = ¯ yθ with θ ∈ Γ + and ¯ x R [ X ] + ¯ y R [ X ] = R [ X ]. Then (2) becomes¯ x/ ¯ y = a ′ /b ′ = c ′ / ( δ − a ′ ) , and, since ¯ x, ¯ y are coprime, there exist t, s ∈ R [ X ] such that a ′ = ¯ xt , b ′ = ¯ yt , c ′ = ¯ xs and δ − a ′ = ¯ ys . Therefore, from the last equality we get ¯ xt + ¯ ys = δ ,i.e., xt + ys = δθ = σ . Moreover, the equality p = p ′ a + q ′ c yields x ′ t + y ′ s = ( ηδθ ) /γ = ζ. The last assumption derives from the fact The fact a, b and c are elements of D ,so deg(¯ xt ) , deg(¯ yt ) , deg(¯ xs ), deg(¯ ys ) are all less or equal than deg δ . (cid:3) Lemma 4.2.
Let x , y be two non-zero weakly comaximal polynomials in R [ X ] suchthat:(1) deg x > deg y ;(2) deg x is even;(3) deg y is odd;(4) y has a unique root u ∈ R such that sgn( x ( u )) = − sgn(l . c . ( x )) .If there exist t, s, x ′ , y ′ ∈ R [ X ] such that xt + ys = σ, (3) x ′ t + y ′ s = ζ, (4) with σ, ζ ∈ Γ + and deg( xt ) , deg( xs ) , deg( yt ) , deg( ys ) ≤ deg( σ ) , then x ′ and y ′ are both non-zero. Moreover, if l . c . ( x ) > (resp. l . c . ( x ) < ), then x ′ / ∈ Γ + (resp. x ′ / ∈ Γ − ).Proof. We firstly observe that since deg x is even, sgn( x ( u )) = − sgn(l . c . ( x )) impliesthere exist x , x ∈ R roots of x such that x < u < x and sgn x = − sgn(l . c . ( x ))in ( x , x ). Moreover, since u is the unique root of y and deg y is odd, then y ( x ) y ( x ) <
0. From the equality (3) and the assumptions on the degrees, weget deg σ = deg( xt ) > deg( ys ) . Therefore, deg t ≥ deg s , deg t is even and, being σ an element of Γ + , the sign ofthe leading coefficient of t is the same as that of the leading coefficient of x . There-fore, since sgn( x ( u ) t ( u )) > t ( u )) = sgn( x ( u )) = − sgn(l . c . ( x )) = − sgn(l . c . ( t )). Then, there exist t , t ∈ R roots of t such that sgn t = − sgn(l . c . ( t )) in( t , t ). Moreover, (3) and y ( x ) y ( x ) < s ( x ) s ( x ) < exists s ∈ ( x , x ) root of s . It is important noting that sgn( x ( s )) = − sgn(l . c . ( x )),therefore from (3) sgn( t ( s )) = − sgn(l . c . ( t ))Now, since t and s are not elements of Γ, it clearly follows from the equality (4)that x ′ and y ′ must be both nonzero. To prove the last assumption of the lemma,suppose that l . c . ( x ) >
0. The case l . c . ( x ) < s ,we get that x ′ ( s ) t ( s ) = ζ ( s ) > t ( s ) is negative, so is x ′ ( s ). Itclearly follows that x ′ / ∈ Γ + . (cid:3) The above lemmas allow us to identify a class of singular dimension 2 matricesover D that do not admit “easy” factorizations into idempotent factors. Proposition 4.3.
Let p and q be two nonzero weakly comaximal elements of D . If deg p > deg q , deg p is even, deg q is odd and q has a unique root u such that p ( u ) has opposite sign to that of the leading coefficient of p , then the matrix (cid:18) p q (cid:19) cannot factor in M ( D ) as (cid:18) p q (cid:19) = (cid:18) p ′
00 0 (cid:19) T or (cid:18) p q (cid:19) = (cid:18) q ′ (cid:19) T , with T = (cid:18) a bc − a (cid:19) idempotent. Moreover, if the leading coefficient of p is positive,then (cid:18) p q (cid:19) is never a product of two idempotent matrices in M ( D ) .Proof. Let p = x/γ and q = y/γ and assume by contradiction that (cid:18) p q (cid:19) = (cid:18) p ′
00 0 (cid:19) (cid:18) a bc − a (cid:19) , with p ′ = x ′ /η ∈ D and a (1 − a ) = bc . By Lemma 4.1 thereexist t, s ∈ R [ X ], σ, ζ ∈ Γ + such that xt + ys = σ,x ′ t = ζ, and deg( xt ) , deg( xs ) , deg( yt ) , deg( ys ) ≤ deg σ . However, since deg x > deg y , deg x is even, deg y is odd and y has a unique root u ∈ R such that sgn( x ( u )) = − sgn(l . c . ( x )), by Lemma 4.2 the second of the above equalities is impossible. Thesame argument shows that (cid:18) p q (cid:19) = (cid:18) q ′ (cid:19) T , for every q ′ ∈ D and T ∈ M ( D )idempotent.Let l . c . ( p ) = l . c . ( x ) > (cid:18) p q (cid:19) is a product oftwo idempotent matrices over D . Then, by Lemma 2.2 (cid:18) p q (cid:19) = (cid:18) q ′ (cid:19) (cid:18) a bc − a (cid:19) ,with q ′ = y ′ /η and a (1 − a ) = bc . By Lemma 4.1 there exist t, s ∈ R [ X ], σ, ζ ∈ Γ + such that xt + ys = σ,ηt + y ′ s = ζ, but the second equality is impossible by the last part of Lemma 4.2. (cid:3) Example.
The easiest example of 2 × D to which the aboveProposition 4.3 applies and that may not factorize as a product of idempotent matrices is (cid:18) ( X − / (1 + X ) X/ (1 + X )0 0 (cid:19) . This fact highlights how it can bedifficult to prove the existence (or the absence) of a factorization into idempotentfactors for a singular matrix over D , even in dimension 2 with elements with “small”degrees. Remark . As recalled in the introduction, Salce and Zanardo conjectured in [11]that every integral domain R satisfying the property (ID ) should be a B´ezoutdomain. Their assumption, motivated by previous results by Laffey [9], Ruitenburg[10] and Bhaskara Rao [1], is sustained by many examples. Unique factorizationdomains, projective-free domains, local domains and PRINC domains (introducedin [11]) turn to be B´ezout whenever they satisfy property (ID ). In support ofthe conjecture, in [3] it is proved that if every singular 2 × R isa product of idempotent matrices, then R is a Pr¨ufer domain such that everyinvertible 2 × R is a product of elementary matrices. Also, interestingexamples of Pr¨ufer non B´ezout domains not satisfying (ID ) were provided. On theother hand, the recent paper [5] raised some doubts on the validity of Salce andZanardo’s hypothesis. In fact, the authors showed that the big family of dimension2 column-row matrices over a real quadratic integer ring O factorize as productsof idempotent matrices, even when O is not a B´ezout domain. Also in the case ofthe minimal Dress ring D of R ( X ), according to the conjecture, we expect that itdoes not satisfy (ID ). However, even if the results in Section 3 seem to suggestthat singular matrices in M ( D ) admit idempotent factorizations they can also beinterpreted as a useful tool to identify a possible candidate as a counterexample, aspointed out in this last section. References [1] K. P. S. Bhaskara Rao. Products of idempotent matrices over integral domains.
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Laura Cossu, Dipartimento di Matematica “Tullio Levi-Civita”, Via Trieste 63 - 35121Padova, Italy
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