Let $R$ be a commutative Noetherian ring. It is shown that the finitely generated $R$-module $M$ with finite Gorenstein dimension is reflexive if and only if $M_{𝔭}$ is reflexive for $𝔭\in \mathrm{Spec}\left(R\right)$ with $\mathrm{depth}\left(R_{𝔭}\right)\le 1$, and ${\text{G-}\text{dim}}_{R_{𝔭}}\left(M_{𝔭}\right)\le \mathrm{depth}\left(R_{𝔭}\right)-2$ for $𝔭\in \mathrm{Spec}\left(R\right)$ with $\mathrm{depth}\left(R_{𝔭}\right)\ge 2$. This gives a generalization of Serre and Samuel’s results on reflexive modules over a regular local ring and a generalization of a recent result due to Belshoff. In addition, for $n\ge 2$ we give a characterization of $n$-Gorenstein rings via Gorenstein dimension of the dual of modules. Finally it is shown...

For many domains R (including all Dedekind domains of characteristic 0 that are not fields or complete discrete valuation domains) we construct arbitrarily large superdecomposable R-algebras A that are at the same time E(R)-algebras. Here "superdecomposable" means that A admits no (directly) indecomposable R-algebra summands ≠ 0 and "E(R)-algebra" refers to the property that every R-endomorphism of the R-module, A is multiplication by an element of, A.

The class of linear (resp. quadratic) mappings over a commutative ring is determined by a set of equation-type relations. For the class of homogeneous polynomial mappings of degree m ≥ 3 it is so over a field, and over a ring there exists a smallest equationally definable class of mappings containing the preceding one. It is proved that generating relations determining that class can be chosen to be strong relations (that is, of the same form over all commutative rings) if{f} m ≤ 5. These relations...

Let $R$ be a commutative ring with non-zero identity. Various properties of multiplication modules are considered. We generalize Ohm’s properties for submodules of a finitely generated faithful multiplication $R$-module (see [8], [12] and [3]).

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.

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.

We characterize prime submodules of $R\times R$ for a principal ideal domain $R$ and investigate the primary decomposition of any submodule into primary submodules of $R\times R.$