On Tue, 4 Mar 2008, Tom Leinster wrote:
Apologies for the previous trivial question. Here is the correct version.
There's still something odd about this question. Requiring the subcategory to contain all endomorphisms of M of course requires it to contain A(M,M) as a monoid. But if you don't require it to be closed under biproducts in A, then presumably you don't require it to contain A(M,M) as a ring. It therefore raises two questions of "pure algebra": What conditions on a monoid (with 0) are needed to ensure that it occurs as the multiplicative monoid of a ring? Given that it does so occur, can there be several different additive group structures making it into a ring? I suspect that a fair amount must be known about these questions, but the only result I know in this area is one which I quoted in "Stone Spaces": for a ring of the form C(X), X a compact Hausdorff space, the multiplicative monoid structure of C(X) is enough to determine the topology of X (and hence the ring structure of C(X)) uniquely. Peter Johnstone
My colleague Walter Mazorchuk has the following question.
Being abelian is a *property* of a category, not extra structure. Given an abelian category A, it therefore makes sense to define a subcategory of A to be an ABELIAN SUBCATEGORY if, considered as a category in its own right, it is abelian. Note that a priori, the inclusion need not preserve sums, kernels etc.
Now let R be a ring and M an R-module. Is there a minimal abelian subcategory of Mod-R containing M and all its endomorphisms? If so, is there a canonical way to describe it?
Any thoughts or pointers to the literature would be welcome. Feel free to assume hypotheses on R (it might be a finite-dimensional algebra etc), or to answer the question for full subcategories only.
Thanks, Tom