Maryam Salimi | Searcharxiv

arXiv2012

Relative Tor functors with respect to a semidualizing module

We consider relative Tor functors built from resolutions described by a semidualizing module C over a commutative noetherian ring R. We show that the bifunctors Tor^{F_CM}_i (-,-) and Tor^{P_CM}_i (-,-), defined using flat-like and projective-like resolutions, are isomorphic. We show how the vanishing of these functors characterizes the finiteness of the homological dimension F_C-pd, and we use this to give a relation between the F_C-pd of a given module and that of a pure submodule. On the other hand, we show that other relations that one may expect to hold similarly, fail in general. In fact, such relations force the semidualizing modules under consideration to be trivial.

math.AC↗

arXiv2007

Comaximal graph of commutative rings

Let $R$ be a commutative ring with identity. Let $Γ(R)$ be a graph with vertices as elements of $R$, where two distinct vertices $a$ and $b$ are adjacent if and only if $Ra+Rb=R$. In this paper we consider a subgraph $Γ_2(R)$ of $Γ(R)$ which consists of non-unit elements. We look at the connectedness and the diameter of this graph. We completely characterize the diameter of the graph $Γ_2(R)\setminus\J(R)$. In addition, it is shown that for two finite semi-local rings $R$ and $S$, if $R$ is reduced, then $Γ(R)\congΓ(S)$ if and only if $R\cong S$.

math.AC↗