A NOTE ON SIMPLE MODULES OVER QUASI-LOCAL RINGS

Matlis showed that the injective hull of a simple module over a commutative Noetherian ring is Artinian. In several recent papers, noncommutative Noetherian rings whose injective hulls of simple modules are locally Artinian have been studied. This property had been denoted by property (). In this paper we investigate, which non-Noetherian semiprimary commutative quasi-local rings (R, m) satisfy property (). For quasi-local rings (R, m) with m3 = 0, we prove a characterization of this property in terms of the dual space of Soc(R). Furthermore, we show that (R, m) satisfies () if and only if its associated graded ring gr(R) does. Given a field F and vector spaces V and W and a symmetric bilinear map β : V × V → W we consider commutative quasi-local rings of the form F × V × W, whose product is given by (λ1, v1, w1)(λ2, v2, w2) = (λ1λ2, λ1v2 + λ2v1, λ1w2 + λ2w1 + β(v1, v2)) in order to build new examples and to illustrate our theory. In particular we prove that a quasi-local commutative ring with radical cube-zero does not satisfy () if and only if it has a factor, whose associated graded ring is of the form F × V × F with V infinite dimensional and β non-degenerated.