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Register a new userOn a new extension of the zero-divisor graph
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In this paper, we introduce a new graph whose vertices are the nonzero zero-divisors of commutative ring $R$ and for distincts elements $x$ and $y$ in the set $Z(R)^{star}$ of the nonzero zero-divisors of $R$, $x$ and $y$ are adjacent if and only if $xy=0$ or $x+yin Z(R)$. we present some properties and examples of this graph and we study his relation with the zero-divisor graph and with a subgraph of total graph of a commutative ring.
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We continue our study of the new extension of zero-divisor graph. We give a complete characterization for the possible diameters of $widetilde{Gamma}(R)$ and $widetilde{Gamma}(R[x_1,dots,x_n])$, we investigate the relation between the zero-divisor graph, the subgraph of total graph on $Z(R)^{star}$ and $widetilde{Gamma}(R)$ and we present some other properties of $widetilde{Gamma}(R)$.
In this article we introduce the zero-divisor graphs $Gamma_mathscr{P}(X)$ and $Gamma^mathscr{P}_infty(X)$ of the two rings $C_mathscr{P}(X)$ and $C^mathscr{P}_infty(X)$; here $mathscr{P}$ is an ideal of closed sets in $X$ and $C_mathscr{P}(X)$ is the aggregate of those functions in $C(X)$, whose support lie on $mathscr{P}$. $C^mathscr{P}_infty(X)$ is the $mathscr{P}$ analogue of the ring $C_infty (X)$. We find out conditions on the topology on $X$, under-which $Gamma_mathscr{P}(X)$ (respectively, $Gamma^mathscr{P}_infty(X)$) becomes triangulated/ hypertriangulated. We realize that $Gamma_mathscr{P}(X)$ (respectively, $Gamma^mathscr{P}_infty(X)$) is a complemented graph if and only if the space of minimal prime ideals in $C_mathscr{P}(X)$ (respectively $Gamma^mathscr{P}_infty(X)$) is compact. This places a special case of this result with the choice $mathscr{P}equiv$ the ideals of closed sets in $X$, obtained by Azarpanah and Motamedi in cite{Azarpanah} on a wider setting. We also give an example of a non-locally finite graph having finite chromatic number. Finally it is established with some special choices of the ideals $mathscr{P}$ and $mathscr{Q}$ on $X$ and $Y$ respectively that the rings $C_mathscr{P}(X)$ and $C_mathscr{Q}(Y)$ are isomorphic if and only if $Gamma_mathscr{P}(X)$ and $Gamma_mathscr{Q}(Y)$ are isomorphic.
The divisor sequence of an irreducible element (textit{atom}) $a$ of a reduced monoid $H$ is the sequence $(s_n)_{nin mathbb{N}}$ where, for each positive integer $n$, $s_n$ denotes the number of distinct irreducible divisors of $a^n$. In this work we investigate which sequences of positive integers can be realized as divisor sequences of irreducible elements in Krull monoids. In particular, this gives a means for studying non-unique direct-sum decompositions of modules over local Noetherian rings for which the Krull-Remak-Schmidt property fails.
Let $A$ be the polynomial algebra in $r$ variables with coefficients in an algebraically closed field $k$. When the characteristic of $k$ is $2$, Carlsson conjectured that for any $mathrm{dg}$-$A$-module $M$, which has dimension $N$ as a free $A$-module, if the homology of $M$ is nontrivial and finite dimensional as a $k$-vector space, then $Ngeq 2^r$. Here we examine a stronger conjecture concerning varieties of square-zero upper triangular $Ntimes N$ matrices with entries in $A$. Stratifying these varieties via Borel orbits, we show that the stronger conjecture holds when $N = 8$ without any restriction on the characteristic of $k$. This result also verifies that if $X$ is a product of $3$ spheres of any dimensions, then the elementary abelian $2$-group of order $4$ cannot act freely on $X$.
Let $mathfrak{a}$ be an ideal of a commutative noetherian (not necessarily local) ring $R$. In the case $cd(mathfrak{a},R)leq 1$, we show that the subcategory of $mathfrak{a}$-cofinite $R$-modules is abelian. Using this and the technique of way-out functors, we show that if $cd(mathfrak{a},R)leq 1$, or $dim(R/mathfrak{a}) leq 1$, or $dim(R) leq 2$, then the local cohomology module $H^{i}_{mathfrak{a}}(X)$ is $mathfrak{a}$-cofinite for every $R$-complex $X$ with finitely generated homology modules and every $i in mathbb{Z}$. We further answer Question 1.3 in the three aforementioned cases, and reveal a correlation between Questions 1.1, 1.2, and 1.3.
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