The present work aims to exploit the interplay between algebraic properties of rings and graph-theoretic structures some associated graphs. We introduce commutatively closed graphs investigate subsets a ring with help graph theory. In particular, we compute diameter matrix Artinian semisimple algebras.

Over commutative graded local artinian rings, examples are constructed of periodic modules of arbitrary minimal period and modules with bounded Betti numbers, which are not eventually periodic. They provide counterexamples to a conjecture of D. Eisenbud, that every module with bounded Betti numbers over a commutative local ring is eventually periodic of period 2 . It is proved however, that the...

We apply set-theoretic methods to study projective modules and their generalizations over transfinite extensions of simple artinian rings R. prove that if R is small, then the Weak Diamond implies projectivity an arbitrary module can be tested at layer epimorphisms

For an arbitrary ring $R$, the zero-divisor graph of $R$, denoted by $Gamma (R)$, is an undirected simple graph that its vertices are all nonzero zero-divisors of $R$ in which any two vertices $x$ and $y$ are adjacent if and only if either $xy=0$ or $yx=0$. It is well-known that for any commutative ring $R$, $Gamma (R) cong Gamma (T(R))$ where $T(R)$ is the (total) quotient ring of $R$. In this...