Let R be a commutative ring with identity and M be a unital R-module. Then M is called a multiplication module provided for every submodule N of M there exists an ideal I of R such that N = IM. Our objective is to investigate properties of prime and semiprime submodules of multiplication modules. Mathematics Subject Classification: 13C05, 13C13

We deal with classes of prime ideals whose associated graded ring is isomorphic to the Rees algebra of the conormal module in order to describe the divisor class group of the Rees algebra and to examine the normality of the conormal module.

let m be an artinian module over the commutative ring a (with nonzero identity) and a p spec a be such that a is a finitely generated ideal of a and am = m. also suppose that h = h where h. = m/ (0: a )for i

The source of a simple kG-module, for a finite p-solvable group G and an algebraically closed field k of prime characteristic p, is an endo-permutation module (see [Pu1] or [Th]). L. Puig has proved, more precisely, that this source must be isomorphic to the cap of an endo-permutation module of the form ⊗ Q/R∈S Ten P Q Inf Q Q/R(MQ/R), where MQ/R is an indecomposable torsion endo-trivial module...

It is proved that a commutative ring is clean if and only if it is Gelfand with a totally disconnected maximal spectrum. It is shown that each indecomposable module over a commutative ring R satisfies a finite condition if and only if R P is an artinian valuation ring for each maximal prime ideal P. Commutative rings for which each indecomposable module has a local endomorphism ring are studied...

If R is a valuation domain of maximal ideal P with a maximal immediate extension of finite rank it is proven that there exists a finite sequence of prime ideals P = L0 ⊃ L1 ⊃ · · · ⊃ Lm ⊇ 0 such that RLj /Lj+1 is almost maximal for each j, 0 ≤ j ≤ m − 1 and RLm is maximal if Lm 6= 0. Then we suppose that there is an integer n ≥ 1 such that each torsion-free R-module of finite rank is a direct s...

In the theory of modules over commutative rings there are several possibilities of defining associated prime ideals. The usual definition of an associated prime ideal p for a module M is that p is the annihilator of an element of M . In [2] §1 exercise 17 a generalization of this notion is given. p is called weakly associated (faiblement associé) to M if p is minimal in the set of the prime ide...

Many observations about coalgebras were inspired by comparable situations for algebras. Despite the prominent role of prime algebras, the theory of a corresponding notion for coalgebras was not well understood so far. Coalgebras C over fields may be called coprime provided the dual algebra C∗ is prime. This definition, however, is not intrinsic it strongly depends on the base ring being a field...