The basic ring of a locally Artinian commutative Grothendieck category

The context is that of a locally Artinian Grothendieck category---a Grothendieck category with a generating set of Artinian objects---which is, moreover, commutative in the sense that it has a generator with commutative endomorphism ring. C. Menini and Orsatti [Arch. Math. (Basel) 49 (1987), no. 6, 484--496; MR0921114 (89a:18012)] showed that any locally Artinian Grothendieck category is equivalent to a full subcategory of modules over a topological ring (R,τ), called the basic ring of the category. The basic ring, as a topological ring, classifies the category up to natural equivalence. In this paper the authors show that the basic ring of a commutative locally Artinian Grothendieck category is the topological product of commutative local Artinian rings. The proof uses the fact that objects of the category have primary decompositions indexed by the simple modules over the associated basic ring, and involves showing that there is a corresponding decomposition of the ring as a topological product.