Modules with cyclic support

Abstract

In this paper we define a class of modules which we call modules with cyclic support 1 and show tha t it has the double centralizer property (DCP). A similar problem has been studied by Snapper [,10] and Feller [J6]. Snapper found a class of completely indecomposable modules over commutative rings and showed it has the DCP and Feller [6] extended this in the non-commutative case. In Section 4 an example is given to exhibit the fact tha t a module with cyclic support is not necessarily completely indecomposable. Besides this, we have proved the following results in Section 3. (i) Every module M R with cyclic support has the DCP. (ii) Indecomposable R-modules with cyclic support are essential Rextensions of the support xR. (iii) I f M R is a right R-module with cyclic support and if x R = x R ' _ N ~ M, where N is a sub-bimodule of MR,R,, and N R is essential in MR, then N~ has cyclic support. Section 1 consists of definitions and some of the known results essential for the understanding of the material of this paper. In Section 2 we show tha t cyclic bimodules have the DCP. More generally, it is shown tha t the external direct sum x R ~N has the DCP, where x R ~ x R ' is an R R ' bimodule and N R is a module such tha t (0 : x)R ~ (0 : N)R. In Section 4 we consider the algebra A of n × n matrices over a field F. We denote by M its n-dimensional representation space. I f R is a subalgebra of A and if E ---EndR(M), then Endg(M) ~ R iff R is equal to its bicommutants in A. Thus the main theorems of the paper apply to the problem of determining when a subring of the n × n matrix ring over a field is equal to its bicommutants. This enables us to obtain in Section 4 some examples and counterexamples. In Example 4.1