ON A GENERALIZATION OF RIGHT DUO RINGS

Abstract

We study the structure of rings whose principal right ideals contain a sort of two-sided ideals, introducing right �-duo as a generaliza- tion of (weakly) right duo rings. Abelian �-regular rings are �-duo, which is compared with the fact that Abelian regular rings are duo. For a right �-duo ring R, it is shown that every prime ideal of R is maximal if and only if R is a (strongly) �-regular ring with J(R) = N∗(R). This result may be helpful to develop several well-known results related to pm rings (i.e., rings whose prime ideals are maximal). We also extend the right �-duo property to several kinds of ring which have roles in ring theory. Throughout this note every ring is associative with identity unless otherwise specified. Given a ring R (possibly without identity), J(R), N∗(R), N ∗ (R), and N(R) denote the Jacobson radical, the prime radical, the upper nilradical (i.e., sum of all nil ideals), and the set of all nilpotent elements in R, respectively. It is well-known that N ∗ (R) ⊆ J(R) and N∗(R) ⊆ N ∗ (R) ⊆ N(R). We use R(x) (R((x))) to denote the polynomial (power series) ring with an indeterminate x over R. Denote the n by n full (resp., upper triangular) matrix ring over R by Matn(R) (resp., Un(R)). Use eij for the matrix unit with (i,j)-entry 1 and else- where 0. Denote {(aij) ∈ Un(R) | the diagonal entries of (aij) are all equal} by Dn(R). rR(−) (resp., lR(−)) is used to denote a right (resp., left) annihi- lator in R. Q denotes the direct product of rings. Z (Zn) denotes the ring of integers (modulo n). 1. Right �-duo rings In this section we introduce the concept of a right π-duo ring as a general- ization of weakly right duo ring, and study the structure of right π-duo rings. Let R be a ring and M be a right R-module. Buhphang and Rege (4) called M semicommutative if mRa = 0 whenever ma = 0 for m ∈ M and a ∈ R. We first consider the condition (∗): If ma = 0 for m ∈ M and a ∈ R, then mRa n = 0 for some n ≥ 1,