In this paper, by using elementary tools of commutative algebra,we prove the persistence property for two especial classes of rings. In fact, thispaper has two main sections. In the first main section, we let R be a Dedekindring and I be a proper ideal of R. We prove that if I1, . . . , In are non-zeroproper ideals of R, then Ass1(Ik11 . . . Iknn ) = Ass1(Ik11 ) [ · · · [ Ass1(Iknn )for all k1,...

et  be a commutative Noetherian ring,  and  two ideals of  and  a finite -module. In this paper, we have studied the vanishing and relative Cohen-Macaulyness of the functor for relative Cohen-Macauly filtered modules with respect to the ideal  (RCMF). We have shown that the for relative Cohen-Macaulay modules holds for any relative Cohen-Macauly module with respect to  with ........

Let M be an Artinian module over the commutative ring A (with nonzero identity) and a p spec A be such that a is a finitely generated ideal of A and aM = M. Also suppose that H = H where H. = M/ (0: a )for i

Let $R$ be an associative ring with unity. An element $x \in R$ is called $\mathbb{Z}G$-clean if $x=e+r$, where $e$ is an idempotent and $r$ is a $\mathbb{Z}G$-regular element in $R$. A ring $R$ is called $\mathbb{Z}G$-clean if every element of $R$ is $\mathbb{Z}G$-clean. In this paper, we show that in an abelian $\mathbb{Z}G$-regular ring $R$, the $Nil(R)$ is a two-sided ideal of $R$ and $\fra...

Let L be a lattice. The annihilating-ideal graph of is simple whose vertex set the all nontrivial ideals and two distinct vertices I J are adjacent if only I∧J=0. In this paper, crosscap graphs lattices with at most four atoms characterized. These characterizations provide classes multipartite graphs, which embedded in Klein bottle.

In this paper, we define the new notion of quasi-prime ideal which generalizes at once both prime ideal and primary ideal notions. Then a natural topology on the set of quasi-prime ideals of a ring is introduced which admits the Zariski topology as a subspace topology. The basic properties of the quasi-prime spectrum are studied and several interesting results are obtained. Specially, it is pro...