Geodesic
Semi $n$-ideals of commutative rings
Czechoslovak Mathematical Journal, Tome 72 (2022) no. 4, pp. 977-988
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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 \hbox {$n$-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 $f$.
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 \hbox {$n$-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 $f$.
DOI : 10.21136/CMJ.2022.0208-21
Classification : 13A15, 13A99
Keywords: semi $n$-ideal; semiprime ideal; $n$-ideal 
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Yetkin Çelikel, Ece; Khashan, Hani A. Semi $n$-ideals of commutative rings. Czechoslovak Mathematical Journal, Tome 72 (2022) no. 4, pp. 977-988. doi: 10.21136/CMJ.2022.0208-21

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