Commutative endomorphism rings.

Question

Let $A$ be a commutative ring, and $M$ an $A$-module. Are there some reasonably general conditions (both on $A$ or $M$) assuring that the ring $\mathrm{End}_A(M)$ is commutative? I am also interested in classes of $A$-modules which have this property. Do they have a name?

Answer 1

From the first hit of a google search

CĂLUGĂREANU, G. and SCHULTZ, P. (2010) Modules with abelian endomorphism rings, Bulletin of the Australian Mathematical Society, 82(1), pp. 99–112

we can see that the problem is not even completely understood for $A=\mathbb Z$:

It is a long standing problem in abelian group theory to describe those groups $G$ whose endomorphism ring $\mathrm{End}(G)$ is commutative and those commutative rings which are isomorphic to the endomorphism ring of an abelian group. Krylov, Mikhalev and Tuganbaev [KMT03] present an authoritative survey of the history of the problems. The known results are of two sorts: if $G$ is torsion, then a complete classification is known for both parts of the problem [KMT03, Corollary 19.3]; if $G$ is torsion–free, then no classification is feasible; instead, there are realization theorems stating that large classes of commutative rings can be realized as $\mathrm{End}(G)$ for some group $G$ [KMT03, Section 29]. Naturally, mixed groups lie between these two extremes.

The purpose of this paper is to generalize the problem by considering modules whose endomorphism ring is ‘almost commutative’. For example, we describe classes of modules whose endomorphism rings have commuting idempotents, or in which one–sided ideals are two–sided.

The Krylov - Mikhalev - Tuganbaev book appears to be available through googlebooks, too, and that looks to be a worthwhile read.

Another interesting hit was this paper

Cox, S. H. Commutative endomorphism rings. Pacific J. Math. 45 (1973), no. 1, 87--91. http://projecteuclid.org/euclid.pjm/1102947710.

which describes a problem of Vasconcelos about rings where the endomorphism rings of each ideal are commutative. I didn't find anything to apply to this problem from that paper, however.