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2 Journal of Algebra Contents lists available at ScienceDirect Journal of Algebra wwwelseviercom/locate/jalgebra Centralizers in endomorphism rings Vesselin Drensky a,,jenőszigeti b,leonvanwyk c a Institute of Mathematics and Informatics, Bulgarian Academy of Sciences, 1113 Sofia, Bulgaria b Institute of Mathematics, University of Miskolc, Miskolc, Hungary 3515 c Department of Mathematical Sciences, Stellenbosch University, P/Bag X1, Matieland 7602, Stellenbosch, South Africa article info abstract Article history: Received 4 December 2008 Availableonline8October2010 Communicated by Efim Zelmanov MSC: 15A30 15A27 16D60 16R10 16S50 16U70 We prove that the centralizer Cenϕ End R M of a nilpotent endomorphism ϕ of a finitely generated semisimple left R-module R M over an arbitrary ring R is the homomorphic image of the opposite of a certain ZR-subalgebra of the full m m matrix algebra M m, wherem is the dimension of kerϕ IfR is a local ring, then we give a complete characterization of Cenϕ and of the containment Cenϕ Cenσ, whereσ is a not necessarily nilpotent element of End R M For a K -linear map A of a finite dimensional vector space over a field K we determine the PIdegree of CenA 2010 Elsevier Inc All rights reserved Keywords: Centralizer Module endomorphism Nilpotent Jordan normal base 1 Introduction If S is a ring or algebra, then the centralizer Cens ={u S us = su} of an element s S is a subring subalgebra of S The aim of this paper is to investigate the centralizer Cenϕ of an element ϕ in the endomorphism ring End R M of a left R-module R M In the case of finite dimensional vector spaces the study of Cenϕ can be reduced to the nilpotent case Thus we focus only on the The authors were partially supported by Grant MI-1503/2005 of the Bulgarian National Science Fund, OTKA of Hungary K61007 and the National Research Foundation NRF of South Africa under Grant UID respectively Any opinion, findings and conclusions or recommendations expressed in this material are those of the authors and therefore the NRF does not accept any liability in regard thereto * Corresponding author addresses: drensky@mathbasbg V Drensky, jenoszigeti@uni-miskolchu J Szigeti, LvW@sunacza L van Wyk /$ see front matter 2010 Elsevier Inc All rights reserved doi:101016/jjalgebra

3 V Drensky et al / Journal of Algebra nilpotent endomorphisms of a finitely generated semisimple R M We note that most of our statements are generalizations of classical linear algebra results about commuting matrices see [2,5,6,8] Following observations about the nilpotent Jordan normal base in Section 2 and other preliminary results in Section 3, we prove in Theorem 39 that Cenϕ is the homomorphic image of the opposite of a certain ZR-subalgebra of the full m m matrix algebra M m over the polynomial ring, where m is the dimension of kerϕ IfR is a local ring, then in Theorem 311 we present Cenϕ as the opposite of a factor of a certain subalgebra of M m TheZR-dimension of Cenϕ is determined when R is local, ZR is a field and R/ JR is finite dimensional over ZR If ϕ is a so-called indecomposable nilpotent element of End R M, then the elements of Cenϕ are described in terms of an appropriate R-generating set of R M in Theorem 41 In particular, if R is commutative, then ψ Cenϕ if and only if ψ is a polynomial expression of ϕ IfR is a local ring, ϕ is nilpotent and σ is an arbitrary element of End R M, thencenϕ Cenσ is equivalent to the existence of a certain R-generating set of R M Theorem 43 In the commutative local case Cenϕ Cenσ if and only if σ is a polynomial expression of ϕ For a nilpotent matrix A M n K over a field K the semisimple component of CenA is determined in Theorem 51 Our proof of Theorem 51 is based on the use of Theorem 311 If p is the maximum number of elementary Jordan matrices of the same size and with the same eigenvalue of a not necessarily nilpotent A, then for the T-ideals of the identities we prove that T CenA T M p K q for a suitable q Hence the PI-degree of CenA is equal to p Since all known results about matrix centralizers are closely connected with the Jordan normal form, it is not surprising that our development depends on the existence of the so-called nilpotent Jordan normal base of a semisimple module with respect to a given nilpotent endomorphism the main theorem of [7] For a version of this paper containing more computational details see [1] 2 The nilpotent Jordan normal base Throughout the paper a ring R means a not necessarily commutative ring with identity, ZR and J = JR denote the center and the Jacobson radical of R, respectively Also, M m R and denote the m m matrix ring and the polynomial ring of the commuting indeterminate z, respectively The ideal z k generated by z k will be considered in the sequel, and z 0 = AsubsetX ={x γ,i γ Γ, 1 i k γ } of a unitary left R-module R M is called a nilpotent Jordan normal base with respect to ϕ End R M if each R-submodule Rx γ,i M is simple, Rx γ,i = M γ Γ,1 i kγ is a direct sum, ϕx γ,i = x γ,i+1, ϕx γ,k γ = x γ,kγ +1 = 0forallγ Γ,1 i k γ, and the set {k γ γ Γ } of integers is bounded Γ is called the set of Jordan blocks and the size of the block γ Γ is the integer k γ Obviously, the existence of a nilpotent Jordan normal base implies that R M is semisimple and ϕ is nilpotent with ϕ n = 0 ϕ n 1, where n = max{k γ γ Γ } Clearly, imϕ = Rx γ,i and kerϕ = γ Γ Rx γ,k γ, γ Γ, 2 i kγ where Γ ={γ Γ k γ 2} The following is one of the main results in [7] 21 Theorem For ϕ End R M the following are equivalent: 1 R M is semisimple and ϕ is nilpotent 2 There exists a nilpotent Jordan normal base of R Mwithrespecttoϕ

4 3380 V Drensky et al / Journal of Algebra Proposition Let ϕ End R M be nilpotent, with R M finitely generated semisimple If {x γ,i γ Γ, 1 i k γ } and {y δ, j δ, 1 j l δ } are nilpotent Jordan normal bases of R Mwithrespecttoϕ, then there exists a bijection π : Γ such that k γ = l πγ for all γ Γ Thus the sizes of the blocks of a nilpotent Jordan normal base are unique up to a permutation of the blocks Proof We apply induction on the index of the nilpotency of ϕ Ifϕ = 0, then we have k γ = l δ = 1for all γ Γ, δ, and γ Γ Rx γ,1 = δ Ry δ,1 = M implies the existence of a bijection π : Γ Krull Schmidt, Kurosh Ore Assume that our statement holds for any R-endomorphism φ : N N with R N a finitely generated semisimple left R-module and φ n 1 = 0 φ n 2 Consider the situation described in the proposition with ϕ n = 0 ϕ n 1 Then imϕ = γ Γ, 2 i kγ Rx γ,i ensures that {x γ,i γ Γ, 2 i k γ } is a nilpotent Jordan normal base of the left R-submodule imϕ of R M with respect to the restricted R-endomorphism ϕ : imϕ imϕ The same holds for {y δ, j δ, 2 j l δ }, where ={δ l δ 2} Sinceφ n 1 = 0 φ n 2 for φ = ϕ imϕ, our assumption gives a bijection π : Γ such that k γ 1 = l πγ 1forallγ Γ Inviewof kerϕ = γ Γ Rx γ,kγ = δ Ry δ,l δ we have Γ = and so Γ \ Γ = \ Thuswehave abijectionπ : Γ \ Γ \ and the natural map π π : Γ Γ \ Γ \ is a bijection with the desired property We call a nilpotent element s of a ring S decomposable if es = se for some idempotent e S with 0 e 1 A nilpotent element which is not decomposable is called indecomposable In the case of finite dimensional vector spaces an indecomposable nilpotent endomorphism is nonderogatory or 1-regular in the sense of [4] 23 Proposition Let ϕ : M M be a non-zero nilpotent R-endomorphism of the semisimple left R-module R M Then the following are equivalent: 1 There is a nilpotent Jordan normal base {x i 1 i n} of R Mwithrespecttoϕ consisting of one block thus Γ =1 for any nilpotent Jordan normal base {x γ,i γ Γ, 1 i k γ } of R Mwithrespecttoϕ 2 ϕ is an indecomposable nilpotent element of the ring End R M 3 R M is finitely generated and ϕ d 1 0, whered= dim R M Proof 1 3 is straightforward 1 2: If ε ϕ = ϕ ε for some idempotent ε End R M with 0 ε 1, then imε im1 ε = M for the non-zero semisimple R-submodules imε and im1 ε of R M, and ϕ : imε imε and ϕ : im1 ε im1 ε Since these restricted R-endomorphisms are nilpotent, we have nilpotent Jordan normal bases of imε and im1 ε with respect to ϕ imε and ϕ im1 ε respectively The union of these two bases gives a nilpotent Jordan normal base of M with respect to ϕ consisting of more than one block, a contradiction the direct sum property of the new base is a consequence of the modularity of the submodule lattice of R M 2 1: Suppose we have a nilpotent Jordan normal base X of R M with respect to ϕ with Γ 2 Fix δ Γ and consider the non-zero ϕ-invariant R-submodules N δ = Rx δ,i and N δ = 1 i k δ γ Γ \{δ}, 1 i kγ Rx γ,i Then M = N δ N δ and for the natural projection ε δ of M onto N δ we have ε δ ε δ = ε δ,0 ε δ 1 and ε δ ϕ = ϕ ε δ

5 V Drensky et al / Journal of Algebra The centralizer of a nilpotent endomorphism Note that ϕ End R M defines a natural left action of on M providing a left -module structure on M Clearly,Cenϕ = End M for the centralizer ZR-subalgebra of End R M Henceforth R M is semisimple and we consider a fixed nilpotent Jordan normal base X ={x γ,i γ Γ, 1 i k γ } M with respect to a given nilpotent ϕ End R M of index n The Γ -copower γ Γ is an ideal of the Γ -direct power ring Γ comprising all elements f = f γ z γ Γ with a finite set {γ Γ f γ z 0} of non-zero coordinates The copower power has a natural, -bimodule structure If f γ z = a γ,1 + a γ,2 z + +a γ,n γ +1z nγ then Φf = γ Γ,1 i kγ a γ,i x γ,i = γ Γ 1 i kγ a γ,i ϕ i 1 x γ,1 = γ Γ f γ z x γ,1 defines a left -module homomorphism Φ : γ Γ M 31 Proposition The function Φ is surjective, γ Γ J[z] +zk γ kerφ and if R is local R/ J is a division ring, equality holds Proof The surjectivity of Φ and the containment γ Γ J[z]+zk γ kerφ are clear When R is local and a γ,i x γ,i = 0 for some 1 i k γ, then a γ,i J Thus f γ z = a γ,1 + a γ,2 z + + a γ,k γ zk γ 1 + a γ,k γ +1z k γ + +a γ,n γ +1z n γ J[z]+z k γ is a consequence of Φf = 0, implying that f γ Γ J[z]+zk γ From now onward we also require that R M be finitely generated, Γ ={1, 2,,m} and to ease readability we assume that k 1 k 2 k m 1 for the block sizes, in which case dim R M = γ Γ k γ and dim R kerϕ = Γ =m for the dimensions composition lengths Now γ Γ = Γ and an element f = f γ z γ Γ of Γ is a 1 m matrix over Wedefinethefollowing subsets of M m : IX = { [ P M m P = pδ,γ z ] and p δ,γ z J[z]+ z k γ for all δ,γ Γ } J[z]+z k 1 J[z]+z k 2 J[z]+z k m J[z]+z k 1 J[z]+z k 2 J[z]+z k m =, J[z]+z k 1 J[z]+z k 2 J[z]+z k m N X = { P M m P = [ pδ,γ z ] and z k δ p δ,γ z J[z]+ z k γ for all δ,γ Γ }, MX = { P M m fp kerφ for all f kerφ } Note that IX and N X are, -sub-bimodules of M m in a natural way For δ,γ Γ let k δ,γ = k γ k δ when 1 k δ < k γ n and k δ,γ = 0otherwise 32 Remark It can be verified that the condition z k δ p δ,γ z J[z]+z k γ in the definition of N X is equivalent to p δ,γ z J[z]+z k δ,γ, and so

6 3382 V Drensky et al / Journal of Algebra N X = J[z]+z k 1 k 2 J[z]+z k 1 k 3 J[z]+z k 2 k 3 J[z]+z k 1 k m J[z]+z k 2 k m J[z]+z k 3 k m 33 Lemma IX l M m is a left ideal, N X M m is a subring, IX N X is an ideal and MX is a ZR-subalgebra of M m If R is a local ring, then N X = MX Proof Since the γ -th column of the matrices in IX comes from a left ideal J[z]+z k γ of, we can see that IX is a left ideal If P, Q N X, thenwehavep δ,τ z J[z]+z k δ,τ and q τ,γ z J[z]+z k τ,γ Thusk δ,τ + k τ,γ k δ,γ implies that p δ,τ zq τ,γ z J[z]+z k δ,γ It follows that PQ N X proving that N X is a subring If P IX and Q N X, thenwehavep δ,τ z J[z]+z k τ and q τ,γ z J[z]+z k τ,γ Since k τ + k τ,γ k γ, it follows that p δ,τ zq τ,γ z J[z]+z k γ ThusPQ IX and IX is an ideal of N X If P, Q MX and f kerφ, thenfp kerφ implies that ΦfPQ = ΦfPQ = 0, whence fpq kerφ follows Thus PQ MX, proving that MX is a ZR-subalgebra of M m If R is a local ring, then Proposition 31 gives that kerφ = γ Γ J[z]+zk γ Nowe δ kerφ, where e δ denotes the vector with z k δ in its δ-coordinate and zeros in all other places If P MX, then e δ P kerφ implies that z k δ p δ,γ z J[z]+z k γ, whence P N X follows If P N X and f kerφ, then p δ,γ z J[z] +z k δ,γ and f δ z J[z] +z k δ Thus k δ + k δ,γ k γ implies that f δ zp δ,γ z J[z]+z k γ, whence fp kerφ and P MX follows 34 Lemma If the center ZR of the ring R is a field such that R/ J is finite dimensional over ZR, then dim ZR N X/IX = [ R/ J : ZR ] k1 + 3k m 1k m Proof Any ZR-base of R/ J naturally leads to a ZR-base of N X/I X, and so the claim is obvious from the above matrix forms of IX and N X The assumption k 1 k 2 k m 1 ensures that UX = { U M m R/ J U =[uδ,γ ] and u δ,γ = 0if1 k δ < k γ } is a block upper triangular subalgebra of M m R/ J The T-ideal of the identities of UX is described in [3] We note that, if k 1 > k 2 > > k m 1, then is an upper triangular matrix algebra 35 Lemma There is a natural ring isomorphism R/ J R/ J R/ J 0 R/ J UX = R/ J 0 0 R/ J N X/ N X zm m + IX = UX which is an R, R-bimodule isomorphism at the same time

7 V Drensky et al / Journal of Algebra Proof If P =[p δ,γ z] is in N X, then it is straightforward to see that there exists a matrix [w δ,γ ] in M m R N X such that P + N X zm m + IX =[wδ,γ ]+ N X zm m + IX holds in N X/N X zm m + IX The assignment P + N X zm m + IX [wδ,γ + J] is well defined and gives the required isomorphism 36 Lemma For P MX and f = f γ z γ Γ in Γ the formula ψ P Φf = ΦfP properly defines an R-endomorphism ψ P : M Mof R Msuchthatψ P ϕ = ϕ ψ P The assignment ΛP = ψ P gives a homomorphism MX op Cenϕ of ZR-algebras Proof Using the definition of M X and the surjectivity of Φ it is straightforward to check the claims 37 Lemma I X kerλ, and if R is local, then the equality holds Proof The containment is clear If R is a local ring and P kerλ, then ΛP = ψ P = 0 implies that ψ P Φf = ΦfP = 0 for all f Γ If 1 δ denotes the vector in Γ with 1 in its δ-coordinate and zeros in all other places, then 1 δ P kerφ and Proposition 31 gives that p δ,γ z J[z]+z k γ 38 Lemma If ψ ϕ = ϕ ψ for some ψ End R M, then there is a P MX such that ψφf = ΦfP for all f = f γ z γ Γ in Γ Proof Since Φ : Γ M is surjective, for each δ Γ we can find an element p δ = p δ,γ z γ Γ in Γ such that Φp δ = ψx δ,1 Forthem m matrix P =[p δ,γ z] we have ψ Φf = ψ f δ z x δ,1 = f δ z ψx δ,1 = δ Γ δ Γ δ Γ f δ z Φp δ = δ Γ Φ f δ zp δ = Φ δ Γ f δ zp δ = ΦfP for all f Γ Sincef kerφ implies that ΦfP = ψφf = 0, we obtain that P MX 39 Theorem Λ : MX op Cenϕ is a surjective homomorphism of ZR-algebras Proof The claim directly follows from Lemma 36 and Lemma Corollary Cenϕ satisfies all the polynomial identities with coefficients in ZR of M op m 311 Theorem If R is a local ring, then Cenϕ is isomorphic to the opposite of the factor N X/IX as ZR-algebras: Cenϕ = N X/IX op = N op X/IX

8 3384 V Drensky et al / Journal of Algebra If f i = 0 are polynomial identities of the ZR-subalgebra U op X of M op m R/ J with f i ZR x 1,,x r, 1 i n, then f 1 f 2 f n = 0 is an identity of Cenϕ Proof Theorem 39 ensures that Cenϕ = MX op / kerλ as ZR-algebras In order to prove the desired isomorphism, it suffices to note that for a local ring R we have MX = N X and kerλ = IX by Lemmas 33 and 37 respectively Thus L = N X zm m + IX /IX N X/IX canbeviewedasanidealofcenϕ The use of Lemma 35 gives Cenϕ/L = N op X/IX /L = N op X/ N X zm m + IX = U op X It follows that f i = 0isanidentityofCenϕ/L Thus f i v 1,,v r L for all v 1,,v r Cenϕ, and so f 1 v 1,,v r f 2 v 1,,v r f n v 1,,v r L n SincezM m n IX implies that L n = {0}, the proof is complete 312 Corollary If R is a local ring such that ZR is a field and R/ J is finite dimensional over ZR, then dim ZR Cenϕ = [ R/ J : ZR ] k1 + 3k m 1k m Proof Since dim ZR N X/IX op = dim ZR N X/IX, the result follows from Lemma 34 4 Further properties of the centralizers 41 Theorem Let ϕ be an indecomposable nilpotent element of End R M Thenψ Cenϕ if and only if there is an R-generating set {y j M 1 j d} of R M and elements a 1, a 2,,a n in R such that for all 1 j d a 1 y j + a 2 ϕy j + +a n ϕ n 1 y j = ψy j and a 1 ϕy j + a 2 ϕ ϕy j + +a n ϕ n 1 ϕy j = ψ ϕy j Proof If ψ Cenϕ, then the first identity implies the second one Proposition 23 ensures the existence of a nilpotent Jordan normal base {x i 1 i n} of R M with respect to ϕ consisting of one block Clearly, 1 i n Rx i = M implies that ψx 1 = a 1 x 1 + a 2 x 2 + +a n x n = a 1 x 1 + a 2 ϕx 1 + +a n ϕ n 1 x 1 for some a 1, a 2,,a n R Thusψx i = ψϕ i 1 x 1 = ϕ i 1 ψx 1 = ϕ i 1 a 1 x 1 + a 2 ϕx a n ϕ n 1 x 1 = a 1 ϕ i 1 x 1 + a 2 ϕϕ i 1 x a n ϕ n 1 ϕ i 1 x 1 = a 1 x i + a 2 ϕx i + +a n ϕ n 1 x i for all 1 i n Conversely, we have ϕψy j = ϕa 1 y j + a 2 ϕy j + +a n ϕ n 1 y j = a 1 ϕy j + a 2 ϕϕy j + +a n ϕ n 1 ϕy j = ψϕy j for all 1 j d Thusϕ ψ = ψ ϕ 42 Corollary If in addition R is commutative, then ψ Cenϕ if and only if there are a 1, a 2,,a n R such that a 1 u + a 2 ϕu + +a n ϕ n 1 u = ψu for all u M, in other words, ψ is a polynomial of ϕ 43 Theorem Let R be a local ring and σ End R M ThenCenϕ Cenσ if and only if there is an R-generating set {y j M 1 j d} of R M and there are elements a 1, a 2,,a n in R such that for all 1 j d and all ψ Cenϕ a 1 ψy j + a 2 ϕ ψy j + +a n ϕ n 1 ψy j = σ ψy j

9 V Drensky et al / Journal of Algebra Proof If Cenϕ Cenσ, thena 1 y j + a 2 ϕy j + +a n ϕ n 1 y j = σ y j implies that a 1 ψy j + a 2 ϕψy j + + a n ϕ n 1 ψy j = σ ψy j for all ψ Cenϕ For any δ Γ we have ε δ Cenϕ Cenσ, where ε δ : M N δ is the natural projection corresponding to the direct sum M = N δ N δ,with N δ = Rx δ,i and N δ = 1 i k δ γ Γ \{δ}, 1 i kγ Rx γ,i Thus σ : imε δ imε δ and so σ x δ,1 = 1 i k δ a δ,i x δ,i = h δ z x δ,1 for some h δ z = a δ,1 + a δ,2 z + +a δ,kδ z kδ 1 in Sinceϕ Cenσ implies that σ Cenϕ, it follows that σ Φf = γ Γ σ f γ z x γ,1 = γ Γ f γ z σ x γ,1 = γ Γ f γ z h γ z x γ,1 = γ Γ f γ zh γ z x γ,1 = ΦfH, where f Γ and H = γ Γ h γ ze γ,γ is a diagonal matrix in MX H MX is a consequence of σ Φf = ΦfH By Theorem 39, the containment Cenϕ Cenσ is equivalent to the condition that σ ψ P = ψ P σ for all P MX As a consequence, we obtain that Cenϕ Cenσ is equivalent to ΦfPH = σ ΦfP = σ ψ P Φf = ψ P σ Φf = ψ P ΦfH = ΦfHP for all f Γ and P MX Thus Cenϕ Cenσ implies ΦfPH HP = 0 or fph HP kerφ Take e = 1 γ Γ and E 1,δ N X by Remark 32 Then the δ-coordinate of ee 1,δ H HE 1,δ = h δ z h 1 zee 1,δ kerφ is h δ z h 1 z SinceR is local, P = E 1,δ MX by the last part of Lemma 33 Now Proposition 31 gives that h δ z h 1 z J[z]+z k δ Thusσ x δ,1 = h δ z x δ,1 = h 1 z x δ,1 for all δ Γ It follows that σ x γ,i = σ ϕ i 1 x γ,1 = ϕ i 1 σ x γ,1 = ϕ i 1 h 1 z x γ,1 = h 1 z ϕ i 1 x γ,1 = h 1 z x γ,i = a 1 x γ,i + a 2 ϕx γ,i + +a n ϕ n 1 x γ,i, where h 1 z = a 1 + a 2 z + +a n z n 1 Conversely, 1 M Cenϕ gives a 1 y j + a 2 ϕy j + +a n ϕ n 1 y j = σ y j for all 1 j d Then ψσ y j = a 1 ψy j +a 2 ϕψy j + +a n ϕ n 1 ψy j = σ ψy j for all ψ Cenϕ and 1 j d Thus ψ σ = σ ψ and so Cenϕ Cenσ 44 Corollary If in addition R is commutative, then Cenϕ Cenσ if and only if there are a 1, a 2,,a n Rsuchthata 1 u + a 2 ϕu + +a n ϕ n 1 u = σ u for all u M, in other words, σ is a polynomial of ϕ 45 Remark Since Cenϕ Cenσ is equivalent to σ CenCenϕ, we may consider Theorem 43 as some kind of double centralizer theorem 5 The centralizer of an arbitrary linear map If K is an algebraically closed field and {λ 1,λ 2,,λ r } is the set of all eigenvalues of A M n K, then CenA is isomorphic to the direct product of the centralizers CenA i, where A i denotes the block diagonal matrix consisting of all Jordan blocks of A having eigenvalue λ i in the diagonal The number of the diagonal blocks in A i is dimkera i λ i I i, andthesizeofa i is d i d i, where d i is the multiplicity of the root λ i in the characteristic polynomial of A SinceCenA i = CenA i λ i I i and A i λ i I i is nilpotent in M di K, we shall consider the case of a nilpotent matrix 51 Theorem If A M d K is nilpotent of index n, then CenA/ JCenA = M q1 K M qn K, where q e is the number of elementary Jordan matrices of size e eandm qe K ={0} if q e = 0 Theindexof nilpotency of JCenA is bounded from above by nv, where v is the number of different sizes Proof Now A End K K d has a nilpotent Jordan normal base X in K d with block sizes n = k 1 k 2 k m 1, and Theorem 311 gives an isomorphism CenA = N op X/IX of K -algebras Let T i = K [z]/z k i, and to minimize the noise in the matrix below, we use z instead of z + z k i in T i for the K -algebra

10 3386 V Drensky et al / Journal of Algebra C A = T 1 z k 1 k 2 T 1 z k 1 k m T 1 T 2 T 2 z k 2 k 3 T 2 z k 2 k m T 2 T m 1 T m 1 T m 1 z k m 1 k m T m 1 T m T m T m T m Thus the map P + IX [p i, j z + z k j ] is well defined and provides an N op X/IX C A isomorphism of K -algebras, where P =[p i, j z] is in N X and denotes the transpose Recall that the Jacobson radical of a finite dimensional algebra is equal to the maximal nilpotent ideal of the algebra The K [z]-module T 1 T 1 T 1 T 2 T 2 T 2 T A = T m T m T m satisfies z k 1T A ={0} The intersection I = zt A C A is an ideal of C A and I n = I k 1 ={0}, thus I JC A We obtain that C A /I is a lower block triangular matrix algebra with diagonal blocks of size q t1 q t1, q t2 q t2,,q tv q tv, where k 1 = t 1 > t 2 > > t v = k m 1 are the different block sizes the strictly decreasing sequence of the different k i s appearing in X The strictly lower triangular part M qt2 q t1 K 0 0 M qt v q t 1 K M qt v q t v 1 K 0 of C A /I is nilpotent of index v and is equal to the radical of C A /I Consequently, JC A v n I n = {0} and the index of nilpotency of JC A is bounded by nv Clearly,C A / JC A = Mqt1 K M qtv K Note that the form of the centralizer C A in Theorem 51 is a classically known object that can be found, for instance, in [2, Chapter VIII, 2, pp ] or in [8] Hence Theorem 51 could have been observed without the results of this paper, even if it is a by-product of our general approach Recall that the PI-degree PIdegS of a PI-algebra S is equal to the maximum p such that the multilinear polynomial identities of S follow from the multilinear polynomial identities of M p K 52 Corollary Let A be an n n matrix over an algebraically closed field K and let p be the maximum number of equal elementary Jordan matrices in the canonical Jordan form of A over the algebraic closure of K Then PIdegCenA = p Proof For a finite dimensional K -algebra S with Jacobson radical J the PI-degree of S is equal to the maximal size of the matrix subalgebras of S/ J Applying Theorem 51 one completes the proof 53 Remark Corollary 52 holds for all fields If K is not algebraically closed, then a detailed argument in [1] shows how the algebraic closure of K can be used Acknowledgments The authors are very thankful to the referee for the suggestions improving the exposition The second and the third named authors wish to thank PN Anh and L Márki for fruitful consultations

11 V Drensky et al / Journal of Algebra References [1] V Drensky, J Szigeti, L van Wyk, Centralizers in endomorphism rings, arxiv: v1 [mathra], 13 October 2009 [2] FR Gantmacher, The Theory of Matrices, Chelsea Publishing Co, New York, 2000 [3] A Giambruno, M Zaicev, Polynomial Identities and Asymptotic Methods, Math Surveys Monogr, vol 122, Amer Math Soc, Providence, RI, 2005 [4] MG Neubauer, BA Sethuraman, Commuting pairs in the centralizers of 2-regular matrices, J Algebra [5] VV Prasolov, Problems and Theorems in Linear Algebra, Transl Math Monogr, vol 134, Amer Math Soc, Providence, RI, 1994 [6] DA Suprunenko, RI Tyshkevich, Commutative Matrices, Academic Press, New York, London, 1968 [7] J Szigeti, Linear algebra in lattices and nilpotent endomorphisms of semisimple modules, J Algebra [8] HW Turnbull, AC Aitken, An Introduction to the Theory of Canonical Matrices, Dover Publications, 2004