On a Class of Semicommutative Rings

In this paper, a generalization of the class of semicommutative rings is inves- tigated. A ring R is called central semicommutative if for any a; b 2 R, ab = 0 implies arb is a central element of R for each r 2 R. We prove that some results on semicommutative rings can be extended to central semicommutative rings for this general settings

On a Class of Semicommutative Modules

Let R be a ring with identity,M a right R-module and S = EndR(M). In this note, we introduce S-semicommutative, S-Baer, S-q.-Baer and S-p.q.-Baer modules. We study the relations between these classes of modules. Also we prove if M is an S-semicommutative module, then M is an S-p.q.-Baer module if and only if M[x] is an S[x]-p.q.-Baer module, M is an S-Baer module if and only if M[x] is an S[x]-Baer module, M is an S-q.-Baer module if and only if M[x] is an S[x]-q.-Baer module

On a Generalization of Semisimple Modules

Let R be a ring with identity. A module MR is called an r-semisimple module if for any right ideal I of R, MI is a direct summand of MR which is a generalization of semisimple and second modules.We investigate when an r-semisimple ring is semisimple and prove that a ring R with the number of nonzero proper ideals ≤4 and J (R) = 0 is r -semisimple. Moreover, we prove that R is an r-semisimple ring if and only if it is a direct sum of simple rings and we investigate the structure of module whenever R is an r-semisimple ring