Articles written in Proceedings – Mathematical Sciences

    • On the Finiteness Properties of Matlis Duals of Local Cohomology Modules

      K Khashyarmanesh F Khosh-Ahang

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      Let 𝑅 be a complete semi-local ring with respect to the topology defined by its Jacobson radical, $\mathfrak{a}$ an ideal of 𝑅, and 𝑀 a finitely generated 𝑅-module. Let $D_R(-):=\mathrm{Hom}_R(-,E)$, where 𝐸 is the injective hull of the direct sum of all simple 𝑅-modules. If 𝑛 is a positive integer such that $\mathrm{Ext}^j_R(R/\mathfrak{a},D_R(H^t_{\mathfrak{a}}(M)))$ is finitely generated for all $t>n$ and all $j\geq 0$, then we show that $\mathrm{Hom}_R(R/\mathfrak{a},D_R(H^n_{\mathfrak{a}}(M)))$ is also finitely generated. Specially, the set of prime ideals in $\mathrm{Coass}_R(H^n_{\mathfrak{a}}(M))$ which contains $\mathfrak{a}$ is finite.

      Next, assume that $(R,\mathfrak{m})$ is a complete local ring. We study the finiteness properties of $D_R(H^r_{\mathfrak{a}}(R))$ where 𝑟 is the least integer 𝑖 such that $H^i_{\mathfrak{a}}(R)$ is not Artinian.

    • A generalization of zero divisor graphs associated to commutative rings


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      Let $R$ be a commutative ring with a nonzero identity element. For a natural number $n$, we associate a simple graph, denoted by $\Gamma^{n}_{R}$, with $R^{n}\backslash\{0\}$ as the vertex set and two distinct vertices $X$ and $Y$ in $R^{n}$ being adjacent if and only if there exists an $n\times n$ lower triangular matrix $A$ over $R$ whose entries on the main diagonal are nonzero and one of the entries on the main diagonal is regular such that $X^{T} AY = 0$ or $Y^{T} AX = 0$, where, for a matrix $B$, $B^{T}$ is the matrix transpose of $B$. If $n = 1$, then $\Gamma^{n}_{R}$ is isomorphic to the zero divisor graph $\Gamma(R)$, and so $\Gamma^{n}_{R}$ is a generalization of $\Gamma(R)$ which is called a generalized zero divisor graph of $R$. In this paper, we study some basic properties of $\Gamma^{n}_{R}$. We also determine all isomorphic classes of finite commutative rings whose generalized zero divisor graphs have genus at most three.

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