A NOTE ON SIMPLE MODULES OVER QUASI-LOCAL RINGS

Authors : Paula A A B CARVALHO, , Christian LOMP, Patrick F SMİTH

Pages : 91-106

Doi:10.24330/ieja.440231

View : 16 | Download : 6

Publication Date : 2018-07-05

Article Type : Research Paper

Abstract :Matlis showed that the injective hull of a simple module over a commutative Noetherian ring is Artinian. In several recent papers, non- commutative Noetherian rings whose injective hulls of simple modules are lo- cally Artinian have been studied. This property had been denoted by property insert ignore into journalissuearticles values();. In this paper we investigate, which non-Noetherian semiprimary commu- tative quasi-local rings insert ignore into journalissuearticles values(R;m); satisfy property insert ignore into journalissuearticles values();. For quasi-local rings insert ignore into journalissuearticles values(R;m); with m3 = 0, we prove a characterization of this property in terms of the dual space of Socinsert ignore into journalissuearticles values(R);. Furthermore, we show that insert ignore into journalissuearticles values(R;m); satises insert ignore into journalissuearticles values(); if and only if its associated graded ring grinsert ignore into journalissuearticles values(R); does. Given a eld F and vector spaces V and W and a symmetric bilinear map : V V ! W we consider commutative quasi-local rings of the form F V W, whose product is given by insert ignore into journalissuearticles values(1; v1;w1);insert ignore into journalissuearticles values(2; v2;w2); = insert ignore into journalissuearticles values(12; 1v2 + 2v1; 1w2 + 2w1 + insert ignore into journalissuearticles values(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 sat- isfy insert ignore into journalissuearticles values(); if and only if it has a factor, whose associated graded ring is of the form F V F with V innite dimensional and non-degenerated.
Keywords : Quasi local ring, injective hull, simple module, niteness condi tion