On Σ-v Rings

We discuss various properties of a ring over which each simple module is Σ-injective. We shall consider associative rings with identity. Our modules will be unital right modules unless stated otherwise. The class of right V rings was introduced by Villamayor [20]. A ring R is called a right V ring if each simple right Rmodule is injective. It is a well-known unpublished result due to Kaplansky that a commutative ring is von Neumann regular if and only if it is a V ring. The class of V rings and, in particular, V domains has been studied by many authors including Baccella ([1], [2]), Cozzens [5], Faith ([7], [8]), Fisher [10], Goodearl [14], Goursaud [12], Huynh et al [16], Kaplansky, Jain et al [17], Michler et al. [20], Osofsky [21], Page [22], Tyukavkin [24] and Tuganbaev [25]. Goursaud and Valette generalized the notion of V ring and called a ring R to be a right Σ-V ring if each simple right R-module is Σ-injective [13]. Recall that a module M is called Σ-injective if every direct sum of copies of M is injective [6]. After defining these rings, Goursaud and Valette studied only the particular class of von Neumann regular right Σ-V rings [13]. Their paper [13] was published in 1975 but surprisingly in past 34 years no other paper except that of Baccella [1] even mentions Σ-V rings and this class of rings seems to have been forgotten. The present paper attempts to bring this interesting class of rings in light again for further research. Clearly, every right Σ-V ring is a right V ring, however there are examples of right V rings which are not right Σ-V rings. 1. Finiteness Properties of Σ-V rings It is well-known that a ring R is right noetherian if and only if each injective right R-module is Σ-injective [3]. Since in a right Σ-V ring we require each simple module to be Σ-injective, we expect some kind of finiteness in such rings. In this section we will establish finiteness properties of Σ-V rings. We begin with a useful observation about Σ-V rings. First, recall that a ring R is called right q.f.d. relative to a module M if no cyclic right R-module contains an infinite direct sum of modules isomorphic to submodules of M . Lemma 1. Let R be a right Σ-V ring and let S be any simple right R-module. Then R is right q.f.d. relative to S. Proof. It follows from the fact that if a right R-module M is Σ-injective, then R must be right q.f.d. relative to M (see [15]). 2000 Mathematics Subject Classification. 16D50, 16P40.