We prove that for a commutative ring $R$, every noetherian (artinian) $R$-module is quasi-injective if and only if every noetherian (artinian) $R$-module is quasi-projective if and only if the class of noetherian (artinian) $R$-modules is socle-fine if and only if the class of noetherian (artinian) $R$-modules is radical-fine if and only if every maximal ideal of $R$ is idempotent.

In this paper we study commutative rings $R$ whose prime ideals are direct sums of cyclic modules. In the case $R$ is a finite direct product of commutative local rings, the structure of such rings is completely described. In particular, it is shown that for a local ring $(R,ℳ)$, the following statements are equivalent: (1) Every prime ideal of $R$ is a direct sum of cyclic $R$-modules; (2) $ℳ={⨁}_{\lambda \in \Lambda }R{w}_{\lambda }$ where $\Lambda$ is an index set and $R/Ann\left(w_{\lambda }\right)$ is a principal ideal ring for each $\lambda \in \Lambda$; (3) Every prime ideal of $R$ is a direct sum of at most...

If $A$ is a domain with the ascending chain condition on (integral) invertible ideals, then the group $I(A)$ of its invertible ideals is generated by the set $I_m(A)$ of maximal invertible ideals. In this note we study some properties of $I_m(A)$ and we prove that, if $I(A)$ is a free group on $I_m(A)$, then $A$ is a locally factorial Krull domain.

Let $R$ be a ring with an identity (not necessarily commutative) and let $M$ be a left $R$-module. This paper deals with multiplication and comultiplication left $R$-modules $M$ having right ${End}_R\left(M\right)$-module structures.

We shall prove that if $M$ is a finitely generated multiplication module and $\mathrm{A}nn\left(M\right)$ is a finitely generated ideal of $R$, then there exists a distributive lattice $\overline{M}$ such that $\mathrm{S}pec\left(M\right)$ with Zariski topology is homeomorphic to $\mathrm{S}pec\left(\overline{M}\right)$ to Stone topology. Finally we shall give a characterization of finitely generated multiplication $R$-modules $M$ such that $\mathrm{A}nn\left(M\right)$ is a finitely generated ideal of $R$.

We investigate some properties of $n$-submodules. More precisely, we find a necessary and sufficient condition for every proper submodule of a module to be an $n$-submodule. Also, we show that if $M$ is a finitely generated $R$-module and $\sqrt{{\mathrm{Ann}}_R\left(M\right)}$ is a prime ideal of $R$, then $M$ has $n$-submodule. Moreover, we define the notion of $G.n$-submodule, which is a generalization of the notion of $n$-submodule. We find some characterizations of $G.n$-submodules and we examine the way the aforementioned notions are related to each...

We find complete sets of generating relations between the elements [r] = rⁿ - r for $n=2^l$ and for n = 3. One of these relations is the n-derivation property [rs] = rⁿ[s] + s[r], r,s ∈ R.

First, we give a complete description of the indecomposable prime modules over a Dedekind domain. Second, if $R$ is the pullback, in the sense of [9], of two local Dedekind domains then we classify indecomposable prime $R$-modules and establish a connection between the prime modules and the pure-injective modules (also representable modules) over such rings.