Primary-like and weakly primary-like submodules are two new generalizations of primary ideals from rings to modules. In fact, the class of primary-like submodules of a module lie between primary submodules and weakly primary-like submodules properly. In this note, we show that these three classes coincide when their elements are submodules of a multiplication module and satisfy the primeful pro...

let $r$ be a commutative ring with identity‎. ‎a proper ideal $p$ of $r$ is a $(n-1,n)$-$phi_m$-prime ($(n-1,n)$-weakly prime) ideal if $a_1,ldots,a_nin r$‎, ‎$a_1cdots a_nin pbackslash p^m$ ($a_1cdots a_nin pbackslash {0}$) implies $a_1cdots a_{i-1}a_{i+1}cdots a_nin p$‎, ‎for some $iin{1,ldots,n}$; ($m,ngeq 2$)‎. ‎in this paper several results concerning $(n-1,n)$-$phi_m$-prime and $(n-1,n)$-...

Let R be a commutative semiring with $ 1 \neq0$ . In this paper, we study the concept of weakly 1-absorbing primary ideal which is generalization overccommutative semirings A proper I called if whenever nonunit elements $a,b,c \in R$ and $0 \neq abc I$, then $ab $or $c \sqrt{I}$. number results concerning ideals examples are given. And need special type k-ideal or subtractive helps to prove rel...

Let [Formula: see text] be a commutative ring with text]. We introduce the concept of weakly 1-absorbing primary ideal, which is generalization ideal. A proper ideal said to if whenever nonunit elements and text], we have or number results concerning ideals are given, as well examples ideals. Furthermore, give corrected version result on rings.

The goal of this article is to present the graded weakly S-primary ideals and g-weakly which are extensions primary ideals. We state P a ideal R if there exists s ∈ S such that for all x, y h(R), 0 6=xy P, then sx or sy Grad(P). Several properties characteristics as well investigated.