More on the strongly 1-absorbing primary ideals of commutative rings

Ali Yassine; Mohammad Javad Nikmehr; Reza Nikandish

Czechoslovak Mathematical Journal (2024)

  • Issue: 1, page 115-126
  • ISSN: 0011-4642

Abstract

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Let R be a commutative ring with identity. We study the concept of strongly 1-absorbing primary ideals which is a generalization of n -ideals and a subclass of 1 -absorbing primary ideals. A proper ideal I of R is called strongly 1-absorbing primary if for all nonunit elements a , b , c ∈ R such that a b c ∈ I , it is either a b ∈ I or c ∈ 0 . Some properties of strongly 1-absorbing primary ideals are studied. Finally, rings R over which every semi-primary ideal is strongly 1-absorbing primary, and rings R over which every strongly 1-absorbing primary ideal is prime (or primary) are characterized. Many examples are given to illustrate the obtained results.

How to cite

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Yassine, Ali, Nikmehr, Mohammad Javad, and Nikandish, Reza. "More on the strongly 1-absorbing primary ideals of commutative rings." Czechoslovak Mathematical Journal (2024): 115-126. <http://eudml.org/doc/299240>.

@article{Yassine2024,
abstract = {Let $R$ be a commutative ring with identity. We study the concept of strongly 1-absorbing primary ideals which is a generalization of $n$-ideals and a subclass of $1$-absorbing primary ideals. A proper ideal $I$ of $R$ is called strongly 1-absorbing primary if for all nonunit elements $a,b,c \in R$ such that $abc \in I$, it is either $ab \in I$ or $c \in \sqrt\{0\}$. Some properties of strongly 1-absorbing primary ideals are studied. Finally, rings $R$ over which every semi-primary ideal is strongly 1-absorbing primary, and rings $R$ over which every strongly 1-absorbing primary ideal is prime (or primary) are characterized. Many examples are given to illustrate the obtained results.},
author = {Yassine, Ali, Nikmehr, Mohammad Javad, Nikandish, Reza},
journal = {Czechoslovak Mathematical Journal},
keywords = {strongly 1-absorbing primary ideal; $n$-ideal; primary ideal; semi-primary ideal},
language = {eng},
number = {1},
pages = {115-126},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {More on the strongly 1-absorbing primary ideals of commutative rings},
url = {http://eudml.org/doc/299240},
year = {2024},
}

TY - JOUR
AU - Yassine, Ali
AU - Nikmehr, Mohammad Javad
AU - Nikandish, Reza
TI - More on the strongly 1-absorbing primary ideals of commutative rings
JO - Czechoslovak Mathematical Journal
PY - 2024
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
IS - 1
SP - 115
EP - 126
AB - Let $R$ be a commutative ring with identity. We study the concept of strongly 1-absorbing primary ideals which is a generalization of $n$-ideals and a subclass of $1$-absorbing primary ideals. A proper ideal $I$ of $R$ is called strongly 1-absorbing primary if for all nonunit elements $a,b,c \in R$ such that $abc \in I$, it is either $ab \in I$ or $c \in \sqrt{0}$. Some properties of strongly 1-absorbing primary ideals are studied. Finally, rings $R$ over which every semi-primary ideal is strongly 1-absorbing primary, and rings $R$ over which every strongly 1-absorbing primary ideal is prime (or primary) are characterized. Many examples are given to illustrate the obtained results.
LA - eng
KW - strongly 1-absorbing primary ideal; $n$-ideal; primary ideal; semi-primary ideal
UR - http://eudml.org/doc/299240
ER -

References

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