arXiv subjects
Publications and source records attributed to N. Shirmohammadi.
In this paper we improve the recent results on the transfer of Prüfer, Gaussian and arithmetical conditions on amalgamated constructions. As an application we provide an answer to a question posed by Chhiti, Jarrar, Kabbaj and Mahdou as well as we construct various examples.
Our goal is to determine when the trivial extensions of commutative rings by modules are Cohen-Macaulay in the sense of Hamilton and Marley. For this purpose, we provide a generalization of the concept of Cohen-Macaulayness of rings to modules.
Let $A$ and $B$ be commutative rings with unity, $f:A\to B$ a ring homomorphism and $J$ an ideal of $B$. Then the subring $A\bowtie^fJ:=\{(a,f(a)+j)|a\in A$ and $j\in J\}$ of $A\times B$ is called the amalgamation of $A$ with $B$ along $J$ with respect to $f$. In this paper, we study the property of Cohen-Macaulay in the sense of ideals which was introduced by Asgharzadeh and Tousi, a general notion of the usual Cohen-Macaulay property (in the Noetherian case), on the ring $A\bowtie^fJ$. Among other things, we obtain a generalization of the well-known result that when the Nagata's idealization is Cohen-Macaulay.
Let $A$ and $B$ be commutative rings with unity, $f:A\to B$ a ring homomorphism and $J$ an ideal of $B$. Then the subring $A\bowtie^fJ:=\{(a,f(a)+j)|a\in A$ and $j\in J\}$ of $A\times B$ is called the amalgamation of $A$ with $B$ along with $J$ with respect to $f$. In this paper, among other things, we investigate the Cohen-Macaulay and (quasi-)Gorenstein properties on the ring $A\bowtie^fJ$.
Some relations between cohomological dimensions and depths of linked ideals are investigated and discussed by various examples.
Let $R=\bigoplus_{n\geq 0}R_n$, $\fa\supseteq \bigoplus_{n> 0}R_n$ and $M$ and $N$ be a standard graded ring, an ideal of $R$ and two finitely generated graded $R$-modules, respectively. This paper studies the homogeneous components of graded generalized local cohomology modules. First of all, we show that for all $i\geq 0$, $H^i_{\fa}(M, N)_n$, the $n$-th graded component of the $i$-th generalized local cohomology module of $M$ and $N$ with respect to $\fa$, vanishes for all $n\gg 0$. Furthermore, some sufficient conditions are proposed to satisfy the equality $\sup\{\en(H^i_{\fa}(M, N))| i\geq 0\}= \sup\{\en(H^i_{R_+}(M, N))| i\geq 0\}$. Some sufficient conditions are also proposed for tameness of $H^i_{\fa}(M, N)$ such that $i= f_{\fa}^{R_+}(M, N)$ or $i= \cd_{\fa}(M, N)$, where $f_{\fa}^{R_+}(M, N)$ and $\cd_{\fa}(M, N)$ denote the $R_+$-finiteness dimension and the cohomological dimension of $M$ and $N$ with respect to $\fa$, respectively. We finally consider the Artinian property of some submodules and quotient modules of $H^j_{\fa}(M, N)$, where $j$ is the first or last non-minimax level of $H^i_{\fa}(M, N)$.
In this paper, after giving a criterion for a Noetherian local ring to be quasi-Gorenstein, we obtain some sufficient conditions for a quasi- Gorenstein ring to be Gorenstein. In the course, we provide a slight generalization of a theorem of Evans and Griffith.