Let $M$ be a free module over a noetherian ring. For ${\omega }_1,...,{\omega }_k\in M$, let $𝒜$ be the ideal generated by coefficients of ${\omega }_1\wedge ...\wedge {\omega }_k$. For an element $\omega \in {\bigwedge }^{p}M$ with $p\<\mathrm{prof}.𝒜$, if $\omega \wedge {\omega }_1\wedge ...\wedge {\omega }_k=0$, there exists ${\eta }_1,...,{\eta }_k\in {\bigwedge }^{p-1}M$ such that $\omega ={\sum }_{i=1}^k{\eta }_i\wedge {\omega }_i$.This is a generalization of a lemma on the division of forms due to de Rham (Comment. Math. Helv., 28 (1954)) and has some applications to the study of singularities.

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...

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 $R\times R.$

The purpose of this paper is to present a new approach to the classification of indecomposable pseudo-prime multiplication modules over pullback of two local Dedekind domains. We extend the definitions and the results given by Ebrahimi Atani and Farzalipour (2009) to more general cases.

Let $R$ be a commutative Noetherian ring, $I$ an ideal of $R$ and $M$ an $R$-module. We wish to investigate the relation between vanishing, finiteness, Artinianness, minimaxness and $𝒞$-minimaxness of local cohomology modules. We show that if $M$ is a minimax $R$-module, then the local-global principle is valid for minimaxness of local cohomology modules. This implies that if $n$ is a nonnegative integer such that ${\left(H_I^i\left(M\right)\right)}_{𝔪}$ is a minimax $R_{𝔪}$-module for all $𝔪\in \mathrm{Max}\left(R\right)$ and for all $i<n$, then the set ${\mathrm{Ass}}_R\left(H_I^n\left(M\right)\right)$ is finite. Also, if $H_I^i\left(M\right)$ is minimax for...

We are extending to linear recurrent codes, i.e., to time-varying convolutional codes, most of the classic structural properties of fixed convolutional codes. We are also proposing a new connection between fixed convolutional codes and linear block codes. These results are obtained thanks to a module-theoretic framework which has been previously developed for linear control.

We are extending to linear recurrent codes, i.e., to time-varying convolutional codes, most of the classic structural properties of fixed convolutional codes. We are also proposing a new connection between fixed convolutional codes and linear block codes. These results are obtained thanks to a module-theoretic framework which has been previously developed for linear control.