Centralizers in endomorphism rings

We prove that the centralizer Cen(f) in Hom_R(M,M) of a nilpotent endomorphism f of a finitely generated semisimple left R-module M (over an arbitrary ring R) is the homomorphic image of the opposite of a certain Z(R)-subalgebra of the full m x m matrix algebra M_m(R[z]), where m is the dimension (composition length) of ker(f). If R is a local ring, then we provide an explicit description of the above Cen(f). If in addition Z(R) is a field and R/J(R) is finite dimensional over Z(R), then we give a formula for the Z(R)-dimension of Cen(f). If R is a local ring, f is as above and g is an arbitrary element of Hom_R(M,M), then we give a complete description of the containment Cen(f) in Cen(g) in terms of an appropriate R-generating set of M. Using our results about nilpotent endomorphisms, for an arbitrary (not necessarily nilpotent) linear map f in Hom_K(V,V) of a finite dimensional vector space V over a field K we determine the PI-degree of Cen(f) and give other information about the polynomial identities of Cen(f)

Involution Matrix Algebras – Identities and Growth

2000 Mathematics Subject Classification: 16R50, 16R10.The paper is a survey on involutions (anti-automorphisms of order two) of different kinds. Starting with the first systematic investigations on involutions of central simple algebras due to Albert the author emphasizes on their basic properties, the conditions on their existence and their correspondence with structural characteristics of the algebras. Focusing on matrix algebras a complete description of involutions of the first kind on Mn(F) is given. The full correspondence between an involution of any kind for an arbitrary central simple algebra A over a field F of characteristic 0 and an involution on Mn(A) specially defined is studied. The research mainly in the last 40 years concerning the basic properties of involutions applied to identities for matrix algebras is reviewed starting with the works of Amitsur, Rowen and including the newest results on the topic. The cocharactes, codimensions and growth of algebras with involutions are considered as well.Partially supported by Grant MM1106/2001 of the Bulgarian Foundation for Scientific Research