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Keywords:
1-absorbing primary ideal; $n$-ideal; weakly $n$-ideal; weakly $(1,n)$-ideal
1-absorbing primary ideal; $n$-ideal; weakly $n$-ideal; weakly $(1,n)$-ideal
Summary:
We study weakly $(1,n)$-ideals and weakly $n$-ideals in commutative rings. Let $A$ be a commutative ring with a nonzero identity and $I$ be a proper ideal of $A$. Then $I$ is said to be a weakly $(1,n)$-ideal (or weakly $n$-ideal) if whenever $0\neq abc\in I$ for some nonunits $a,b,c\in A$ (or $0\neq ab\in I$ for some $a,b\in A)$, then either $ab\in I$ or $c\in \mathfrak {N}(A)$ (or $a\in I$ or $b\in \mathfrak {N}(A)$, respectively), where $\mathfrak {N}(A)$ is the set of all nilpotent elements of $A$. Many examples and properties of weakly $(1,n)$-ideals and weakly $n$-ideals are given. We characterize all rings in which every proper ideal is a weakly $(1,n)$-ideal and weakly $n$-ideal. Furthermore, we investigate both weakly $(1,n)$-ideals and weakly $n$-ideals in amalgamated algebras along an ideal.
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