Semisimple maximal quotient rings

Abstract

Notation and Introduction. R denotes an arbitrary associative ring. A right R-module A over R will be denoted AR. BR is a large submodule of AR (AR is an essential extension of BR), if BR is a submodule of AR having nonzero intersection with every nonzero submodule of AR. A right ideal I of R is a large right ideal, if IR is a large submodule of RR. Given AR, Z(AR) is the singular submodule of AR [9], which consists of all those elements of A whose annihilators in R are large right ideals. Following Johnson [9], Q is a right quotient ring of R if Q is a ring with identity containing R as a subring (the identity of Q is the identity of R if R has one) and RR is a large submodule of QR. The quotient rings considered by Goldie in [6], [7] will be called classical quotient rings. Q is a classical right quotient of R if every regular element (nonzero divisor) of R is a unit in Q and every element of Q is of the form ab -1, a, b E R, b regular in R. In general, a ring R need not possess a classical right quotient ring. Goldie [7], has given necessary and sufficient conditions that a ring possess a classical right quotient ring which is semisimple. Here semisimple means semisimple with minimum condition [8]. This paper is concerned with the question of characterizing those rings which have a semisimple maximal right quotient ring [4], [9], [10], [11] and in this case generalizing some simple well-known results about commutative integral domains, their quotient rings and modules over these domains. Johnson [9] has shown that R has a regular maximal right quotient ring Q if and only if Z(RR) = 0, where Q is a regular ring [13] if every finitely generated right (left) ideal of Q is generated by an idempotent. In this case QR is injective [3] as a right R-module, hence the injective hull of R [2]. A ring R has a semisimple maximal right quotient ring Q if and only if Z(RR) =O and dim RR is finite, where a right R-module M is of finite dimension if every direct sum of submodules of M has only finitely many nonzero summands. This is the main result of ?1. In addition another characterization is given for rings which possess a semisimple classical right quotient ring, namely, R has a semisimple classical right quotient ring if and only if Z(RR) = 0 and if I is a large right ideal of R, then there is an element a E I such that aR is a large right ideal of R. If R has a semisimple classical right quotient ring Q, then it is known [3], that