Institute of Mathematics

Abstract:  Let $R$ be a commutative ring with identity. A proper ideal $I$ is said to be an $n$-ideal of $R$ if for $a,b\in R$, $ab\in I$ and $a\notin\sqrt0$ imply $b\in I$. We give a new generalization of the concept of $n$-ideals by defining a proper ideal $I$ of $R$ to be a semi $n$-ideal if whenever $a\in R$ is such that $a^2\in I$, then $a\in\sqrt0$ or $a\in I$. We give some examples of semi $n$-ideal and investigate semi $n$-ideals under various contexts of constructions such as direct products, homomorphic images and localizations. We present various characterizations of this new class of ideals. Moreover, we prove that every proper ideal of a zero dimensional general ZPI-ring $R$ is a semi $n$-ideal if and only if $R$ is a UN-ring or $R\cong F_1\times F_2\times\cdots\times F_k$, where $F_i$ is a field for $i=1,\dots,k$. Finally, for a ring homomorphism $f R\rightarrow S$ and an ideal $J$ of $S$, we study some forms of a semi $n$-ideal of the amalgamation $R\bowtie^fJ$ of $R$ with $S$ along $J$ with respect to $f$.