Semisecond Modules

Semisecond Modules

Let R be an arbitrary ring. A non-zero R-module M is called a semisecond module if the annihilator of any non-zero homomorphic image of M is semiprime. We study and investigate this notion over noncommutative rings. It is shown that a non-zero R-module M is a semisecond module if and only if for every ideal I of R, MI2 = MI. Semisecondness is a Morita invariant property. We show that if M is an Artinian non-zero R-module, then every non-zero submodule of M has only a finite number of maximal semisecond submodule. Also a non-zero R-module M is Semisecond if and only if for every proper submodule K of M, there exists a semiprime ideal I of R contained in annR(M/K) such that M/K cogenerates R/I.