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.

Let N be a submodule of a module M and a minimal primary decomposition of N is known. A formula to compute Baer's lower nilradical of N is given. The relations between classical prime submodules and their nilradicals are investigated. Some situations in which semiprime submodules can be written as finite intersection of classical prime submodule are stated.

Let R be a commutative ring with identity and let M be a unitary R -module. Then we study the properties of prime and semiprime submodules and consider the connection between some semiprime submodules of M and the quotient module M s (a subset S of a ring R is said to be multiplicatively closed if whenever s, s´ belong to S, then their product ss´ also belongs to S). Also we prove some results, which have been already proved when the modules are free, concerning the prime submodules of any arbitrary R module.

Let R be a commutative ring with identity. A proper submod-ule N of an R-module M is an n-submodule if rm 2 N (r 2 R; m 2 M) with r =2 p AnnR(M), then m 2 N. A number of results concerning n-submodules are given. For example, we give other characterizations of n-submodules. Also various properties of n-submodules are considered.

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.