Let R be a commutative ring with identity and M a unitary R-module. The concept of prime submodule of an R-module M has been discussed. In this paper, we investigate the concept of semiprime submodule of an R-module M. We also, consider relation between the semiprime submodule N of an r-module M and the semiprime IDEAL (N: M) of R and we focus our attention on relation between E (N) and N.

In this paper, we study the notions of endo-semiprime and endo-cosemiprime modules and obtain some related results. For instance, we show that in a right self-injective ring R, all nonzero IDEALs of R are endo-semiprime as right (left) R-modules if and only if R is semiprime. Also, we prove that both being endo-semiprime and being endo-cosemiprime are Morita invariant properties.

The main purpose of this article is to classify all indecomposable quasi-semiprime comultiplication modules over pullback rings of two Dedekind domains and establish a connection between the quasi-semiprime comultiplication modules and the pure-injective modules over such rings. First, we introduce and study the notion of quasi-semiprime comultiplication modules and classify quasi-semiprime comultiplication modules over local Dedekind domains. Second, we get all indecomposable separated quasi-semiprime comultiplication modules and then, using this list of separated quasi-semiprime comultiplication modules, we show that non-separated indecomposable quasi-semiprime comultiplication Rmodules with finite-dimensional top are factor modules of finite direct sums of separated indecomposable quasi-semiprime comultiplication R-modules.

Let R be a 2-torsion free semiprime ring with extended centroid C, U the Utumi quotient ring of R and m; n>0 are xed integers. We show that if R admits derivation d such that b [[d (x); x]n; [y; d (y)]m]=0 for all x; y Î R where 0¹bÎR, then there exists a central idempotent element e of U such that eU is commutative ring and d induce a zero derivation on (1-e) U. We also obtain some related result in case R is a non-commutative Banach algebra and d continuous or spectrally bounded.

Let R be a 2-torsion free ring and L a Lie IDEAL of R. An additive mapping F:R®R is called a generalized derivation on R if there exists a derivation d:R®R such that F(xy)=F(x)y+xd(y) holds for all x, yÎR. In the present paper we describe the action of generalized derivations satisfying several conditions on Lie IDEALs of semiprime rings.

We know that every Jordan generalized k-derivation of a é-ring is not a generalized k-derivation of the same, in general. Here, we develop a number of lemmas relating to these derivations of certain é-rings and we show that under some conditions every Jordan generalized k-derivation of a 2-torsion free semiprime é-N ring is a generalized k-derivation.

Let $R$ be a $ 2$-torsion-free semiprime ring and $\theta$ be an epimorphism of $R$ . In this paper , under special hypotheses , we prove that if $T : R\longrightarrow R$ is an additive mapping such that $ $ T(xyx)=θ(x)T(y)θ(x) , $ $ holds for all $x , y\in R$ , then $T$ is a $θ$-centralizer either $R$ is unital or $θ(Z(R))=Z(R)$.