Let $A$ be a Noetherian ring, $I$ be an ideal of $A$ and $sigma$ be a semi-prime operation, different from the identity map on the set of all ideals of $A$. Results of Essan proved that the sets of associated prime ideals of $sigma(I^n)$, which denoted by $Ass(A/sigma(I^n))$, stabilize to $A_{sigma}(I)$. We give some properties of the sets $S^{sigma}_{n}(I)=Ass(A/sigma(I^n))setminus A_{sigma}(I...

It is well-known that the sum of two $z$-ideals in $C(X)$ is either $C(X)$ or a $z$-ideal. The main aim of this paper is to study the sum of strongly $z$-ideals in ${mathcal{R}} L$, the ring of real-valued continuous functions on a frame $L$. For every ideal $I$ in ${mathcal{R}} L$, we introduce the biggest strongly $z$-ideal included in $I$ and the smallest strongly $z$-ideal containing ...

let $m$ be a module over a commutative ring $r$ and let $n$ be a proper submodule of $m$. the total graph of $m$ over $r$ with respect to $n$, denoted by $t(gamma_{n}(m))$, have been introduced and studied in [2]. in this paper, a generalization of the total graph $t(gamma_{n}(m))$, denoted by $t(gamma_{n,i}(m))$ is presented, where $i$ is an ideal of $r$. it is the graph with all elements of $...

assume that $mathbb{d}$ is the open unit disk. applying ozaki's conditions, we consider two classes of locally univalent, which denote by $mathcal{g}(alpha)$ and $mathcal{f}(mu)$ as follows begin{equation*}  mathcal{g}(alpha):=left{fin mathcal{a}:mathfrak{re}left( 1+frac{zf^{prime prime }(z)}{f^{prime }(z)}right)

Let $G$ be a finite group and $Gamma(G)$ the prime graph of $G$‎. ‎Recently people have been using prime graphs to study simple groups‎. ‎Naturally we pose a question‎: ‎can we use prime graphs to study almost‎ ‎simple groups or non-simple groups? In this paper some results in‎ ‎this respect are obtained and as follows‎: ‎$Gcong S_p$ if and only‎ ‎if $|G|=|S_p|$ and $Gamma(G)=Gamma(S_p)$‎, ‎whe...