Rings with quasi-continuous right ideals

Abstract

Abstract. Rings in which each right ideal is quasi-continuous (right %-rings) are shown to be a direct sum of semisimple artinian square full ring and a right square free ring. Among other results it is also shown that (i) a nonlocal right continuous indecomposable right %-ring is either simple artinian or a ring of matrices of a certain type, and (ii) an indecomposable non-local right continuous ring is both a right and a left %-ring if and only if it is a right q-ring. In particular, a non local indecomposable right q-ring is a left q-ring. [7]). The purpose of our paper is to extend this line of research by studying rings in which every right ideal is quasi-continuous (right %-rings). In Section 2 we show that a %-ring is a direct sum of a semisimple artinian square full ring and a square free ring.In Section 3 we study right continuous right %-rings. We show that a non-local, right continuous, indecomposable right %-ring R is either simple artinian or a ring of matrices of a certain type (Theorem 3.8). We show that an indecomposable, nonlocal, right continuous ring R is both a right and a left %-ring if and only if R is a right q-ring (Theorem 3.13). In particular, under these hypotheses, R is a right q-ring if and only if it is a left q-ring.1. De®nitions and preliminaries. Throughout the paper R will be a ring with identity and all R-modules will be unital right R-modules, unless otherwise stated. For modules M, N, the notations N & H M and N & È M respectively serve to denote that N is an essential submodule of M and that N is a direct summand of M. By ZM, oM we denote the singular submodule and socle of M, respectively. JR is the Jacobson Radical of R.M (or EM) stands for the injective hull of M. rom R MY N stands for the set of R-homomorphisms from M to N. ind R M is the set of R-endomorphisms of M. If N & M, a closure of N in M is a maximal essential extension of N in M. N is said to be closed in M if N has no proper essential extension on M. Given right R-modules M and N, M is said to be N-injective i for any K & N, every P rom R KY M is the restriction of some P rom R NY M.For a module M we consider the following conditions.

Prime Goldie Rings of Uniform Dimension at Least Two and with All One-Sided Ideals CS Are Semihereditary

“…Byrd, 1979;Ivanov, 1972Ivanov, , 1996Jain et al, 1969Jain et al, , 1999 This paper is about rings for which finitely generated right ideals are CS. Byrd, 1979;Ivanov, 1972Ivanov, , 1996Jain et al, 1969Jain et al, , 1999 This paper is about rings for which finitely generated right ideals are CS.…”

Rings with each right ideal automorphism-invariant

“…In [25] Jain, Singh and Srivastava studied rings whose each right ideal is a finite direct sum of quasi-injective right ideals and called such rings right Σ-q rings. Jain, López-Permouth and Syed in [22] studied rings with each right ideal quasi-continuous and in [7] Clark and Huynh studied rings with each right ideal, a direct sum of quasi-continuous right ideals.…”