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...

In this paper, we characterize w -Noetherian rings and SM rings. More precisely, in terms of the id="M3"> u -operation on a commutative ring id="M4"> R , prove that id="M5"> is id="M6"> if only direct limit id="M7"> r GV -torsion-free injective id="M8"> -modules id="M9"> SM, whic...

In this note, we study commutative Noetherian local rings having finitely generated modules of finite Gorenstein injective dimension. In particular, we consider whether such rings are Cohen-Macaulay.

In this paper, we investigate the change of rings theorems for the Gorenstein dimensions over arbitrary rings. Namely, by the use of the notion of strongly Gorenstein modules, we extend the well-known first, second, and third change of rings theorems for the classical projective and injective dimensions to the Gorenstein projective and injective dimensions, respectively. Each of the results est...

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...

We obtain some methods to construct a (strongly) proper resolution (resp. coproper coresolution) of one end term in a short exact sequence from that of the other two terms. By using this method, we prove that for a left and right Noetherian ring R, RR satisfies the Auslander condition if and only if so does every flat left R-module, if and only if the injective dimension of the ith term in a mi...