In analogy with the notion of the composite semi-valuations, we define the composite $G$-valuation $v$ from two other $G$-valuations $w$ and $u$. We consider a lexicographically exact sequence $(a,\beta ):A_u\to B_v\to C_w$ and the composite $G$-valuation $v$ of a field $K$ with value group $B_v$. If the assigned to $v$ set $R_v=\{x\in K/v\left(x\right)\ge 0$ or $v\left(x\right)$ non comparable to $0\}$ is a local ring, then a $G$-valuation $w$ of $K$ into $C_w$ is defined with its assigned set $R_w$ a local ring, as well as another $G$-valuation $u$ of a residue field is defined with $G$-value group $A_u$.

Let G be a group with identity e. Let R be a G-graded commutative ring and M a graded R-module. In this paper we will obtain some results concerning the graded comultiplication modules over a commutative graded ring.

Let Γ(R) be the zero divisor graph for a commutative ring with identity. The k-domination number and the 2-packing number of Γ(R), where R is an Artinian ring, are computed. k-dominating sets and 2-packing sets for the zero divisor graph of the ring of Gaussian integers modulo n, Γ(ℤₙ[i]), are constructed. The center, the median, the core, as well as the automorphism group of Γ(ℤₙ[i]) are determined. Perfect zero divisor graphs Γ(R) are investigated.

This article is a brief survey of the recent results obtained by several authors on Hochschild homology of commutative algebras arising from the second author’s paper [21].

The Dedekind-Mertens lemma relates the contents of two polynomials and the content of their product. Recently, Epstein and Shapiro extended this lemma to the case of power series. We review the problem with a special emphasis on the case of power series, give an answer to a question posed by Epstein-Shapiro and investigate extensions of some related results. This note is of expository character and discusses the history of the problem, some examples and announces some new results.

We provide several characterizations and investigate properties of Prüfer modules. In fact, we study the connections of such modules with their endomorphism rings. We also prove that for any Prüfer module M, the forcing linearity number of M, fln(M), belongs to {0,1}.

In this paper, by using the Composition-Diamond lemma for non-associative algebras invented by A. I. Shirshov in 1962, we give Gröbner-Shirshov bases for free Pre-Lie algebras and the universal enveloping non-associative algebra of an Akivis algebra, respectively. As applications, we show I. P. Shestakov’s result that any Akivis algebra is linear and D. Segal’s result that the set of all good words in $X^{**}$ forms a linear basis of the free Pre-Lie algebra $\mathrm{PLie}\left(X\right)$ generated by the set $X$. For completeness,...

We introduce and study a new class of ring extensions based on a new formula involving the heights of their primes. We compare them with the classical altitude inequality and altitude formula, and we give another characterization of locally Jaffard domains, and domains satisfying absolutely the altitude inequality (resp., the altitude formula). Then we study the extensions R ⊆ S where R satisfies the corresponding condition with respect to S (Definition 3.1). This leads to a new characterization...

In this article, we compare different types of representations for series with coefficients in complete idempotent semirings. Each of these representations was introduced to solve a particular problem. We show how they are or are not included one in the other and we present a common generalization of them.

Seguendo le idee presentate nei lavori [1] e [2] si studiano le proprietà dei gruppi di $i$-omotopia per moduli ed omomorfismi di moduli.

Let $R$ be a commutative ring. The annihilator graph of $R$, denoted by $\mathrm{AG}\left(R\right)$, is the undirected graph with all nonzero zero-divisors of $R$ as vertex set, and two distinct vertices $x$ and $y$ are adjacent if and only if ${\mathrm{ann}}_R\left(xy\right)\ne {\mathrm{ann}}_R\left(x\right)\cup {\mathrm{ann}}_R\left(y\right)$, where for $z\in R$, ${\mathrm{ann}}_R\left(z\right)=\{r\in R:rz=0\}$. In this paper, we characterize all finite commutative rings $R$ with planar or outerplanar or ring-graph annihilator graphs. We characterize all finite commutative rings $R$ whose annihilator graphs have clique number $1$, $2$ or $3$. Also, we investigate some properties of the annihilator...

Let $R$ be a commutative Noetherian ring, $𝔞$ an ideal of $R$, $M$ an $R$-module and $t$ a non-negative integer. In this paper we show that the class of minimax modules includes the class of $\mathrm{𝒜ℱ}$ modules. The main result is that if the $R$-module ${\mathrm{Ext}}_R^t(R/𝔞,M)$ is finite (finitely generated), $H_{𝔞}^i\left(M\right)$ is $𝔞$-cofinite for all $i<t$ and $H_{𝔞}^t\left(M\right)$ is minimax then $H_{𝔞}^t\left(M\right)$ is $𝔞$-cofinite. As a consequence we show that if $M$ and $N$ are finite $R$-modules and $H_{𝔞}^i\left(N\right)$ is minimax for all $i<t$ then the set of associated prime ideals of the generalized local cohomology module...