Idempotent rings which are equivalent to rings with identity

Abstract

Let A be a ring such that A=A2, but which does not necessarily have an identity element. In studying properties of the ring A through properties of its modules, it is pointless to consider the category A-MOD of all the left Amodules: for instance, every abelian group -with trivial multiplicationis in y-i-MOD. The natural choice for an interesting category of left /1-modules seems to be the following: if a left A-module AM is unital when AM=M, and is A-torsionfree when the annihilator \m(A) is zero, then yl-mod will be the fullsubcategory of ^4-MOD whose objects are the unital and ^4-torsionfree left ^4-modules. The category .4-mod appears in a number of papers (for instance, [7-9]) and when A has local units [1, 2] or is a left s-unital ring [6, 12], then the objects of A-mod are the unital left A-moduies. A-mod is a Grothendieck category and we study here the question of finding necessary and sufficient conditions on the ring A for ^4-mod to be equivalent to a category Z?-mod of modules over a ring with 1. This was already considered for rings with local units in [1], [2] or [3], and for left s-unital rings in [6]. Our situation is therefore more general. In this paper, all rings will be associative rings, but we do not assume that they have an identity. A ring A has local units [2] when for every finite family au ・・■,an of elements of A there is an idempotent e<BA such that ea,― dj―aje for all j=l, ■・■, n. A left A-module M is said to be unital if M has a spanning set (that is, if AM=M); and M has a finite spanning set when M = y£Axi for a finite family of elements xh ■■■, xn of M. The module AM will be called ^4-torsionfreewhen iM(A)=0. A ring A is said to be left nondegenerate if the left module AA is yl-torsionfree,and A is nondegenerate when it is both left and right nondegenerate (see [10, p. 88]). Clearly, a ring with local units is nondegenerate. The ring A will be called (left) s-unital[12] in case for each a^A (equivalentiy, for every finitefamily au ■■■, an of elements