Semi $n$-ideals of commutative rings

@article{YetkinÇelikel2022,
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 $a\notin \sqrt\{0\}$ 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 $a\in \sqrt\{0\}$ 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 $f\colon R\rightarrow S$ and an ideal $J$ of $S$, we study some forms of a semi $n$-ideal of the amalgamation $R\bowtie ^\{f\}J$ of $R$ with $S$ along $J$ with respect to $f$.},
author = {Yetkin Çelikel, Ece, Khashan, Hani A.},
journal = {Czechoslovak Mathematical Journal},
keywords = {semi $n$-ideal; semiprime ideal; $n$-ideal},
language = {eng},
number = {4},
pages = {977-988},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {Semi $n$-ideals of commutative rings},
url = {http://eudml.org/doc/298907},
volume = {72},
year = {2022},
}

TY - JOUR
AU - Yetkin Çelikel, Ece
AU - Khashan, Hani A.
TI - Semi $n$-ideals of commutative rings
JO - Czechoslovak Mathematical Journal
PY - 2022
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 72
IS - 4
SP - 977
EP - 988
AB - 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 $a\notin \sqrt{0}$ 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 $a\in \sqrt{0}$ 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 $f\colon R\rightarrow S$ and an ideal $J$ of $S$, we study some forms of a semi $n$-ideal of the amalgamation $R\bowtie ^{f}J$ of $R$ with $S$ along $J$ with respect to $f$.
LA - eng
KW - semi $n$-ideal; semiprime ideal; $n$-ideal
UR - http://eudml.org/doc/298907
ER -