On 2-absorbing ideals of commutative semirings

Behzadipour, H.; Nasehpour, Peyman

doi:10.1142/s0219498820500346

J. Algebra Appl.

2019

DOI: 10.1142/s0219498820500346

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On 2-absorbing ideals of commutative semirings

H. Behzadipour

Peyman Nasehpour

Abstract: In this paper, we investigate 2-absorbing ideals of commutative semirings and prove that if a is a nonzero proper ideal of a subtractive valuation semiring S then a is a 2-absorbing ideal of S if and only if a = p or a = p 2 where p = √ a is a prime ideal of S. We also show that each 2-absorbing ideal of a subtractive semiring S is prime if and only if the prime ideals of S are comparable and if p is a minimal prime over a 2-absorbing ideal a, then am = p, where m is the unique maximal ideal of S.2010 Mathemat… Show more

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Cited by 3 publications

References 18 publications

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Some remarks on the comparability of ideals in semirings

Behzadipour

Nasehpour

2022

Asian-European J. Math.

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A semiring is uniserial if its ideals are totally ordered by inclusion. First, we show that a semiring [Formula: see text] is uniserial if and only if the matrix semiring [Formula: see text] is uniserial. As a generalization of valuation semirings, we also investigate those semirings whose prime ideals are linearly ordered by inclusion. For example, we prove that the prime ideals of a commutative semiring [Formula: see text] are linearly ordered if and only if for each [Formula: see text], there is a positive integer [Formula: see text] such that either [Formula: see text] or [Formula: see text]. Then, we introduce and characterize pseudo-valuation semidomains. It is shown that prime ideals of pseudo-valuation semidomains and also of the divided ones are linearly ordered.

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Some remarks on the comparability of ideals in semirings

Behzadipour

Nasehpour

2022

Asian-European J. Math.

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On $n$-absorbing prime ideals of commutative rings

Issoual

Mahdou

Moutui

2022

Hacettepe Journal of Mathematics and Statistics

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This paper investigates the class of rings in which every n-absorbing ideal is a prime ideal, called n-AB ring, where n is a positive integer. We give a characterization of an n-AB ring. Next, for a ring R, we study the concept of Ω(R) = {ω R (I); I is a proper ideal of R}, where ω R (I) = min{n; I is an n-absorbing ideal of R}. We show that if R is an Artinian ring or a Prüfer domain, then Ω(R) ∩ N does not have any gaps (i.e., whenever n ∈ Ω(R) is a positive integer, then every positive integer below n is also in Ω(R)). Furthermore, we investigate rings which satisfy property (**) (i.e., rings R such that for each proper idealwhere M in R (I) denotes the set of prime ideals of R minimal over I). We present several properties of rings that satisfy condition (**). We prove that some open conjectures which concern n-absorbing ideals are partially true for rings which satisfy condition (**). We apply the obtained results to trivial ring extensions.

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On 1-Absorbing Fuzzy Ideals of Commutative Semirings

MEHMET ÖZKAN,

ONAR,

OZKAN

et al. 2023

Journal of Engineering Technology and Applied Sciences

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In this study, our goal is to first examine the algebraic structure of 1-absorbing ideals and apply them on fuzzy sets, and to examine the relationships and algebraic properties between them. In this work, the contribution to the literature, the 1-absorbing fuzzy primary ideal is be examined. We look at some of the features of 1-absorbing fuzzy primary ideals. For instance, we show that I is a 1-absorbing fuzzy primary ideal of P if I is a fuzzy primary ideal of P and I is a 2-absorbing fuzzy primary ideal of P if I is a 1-absorbing fuzzy primary ideal of P. After that, we explain these theorems with instances.

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On 2-absorbing ideals of commutative semirings

Cited by 3 publications

References 18 publications

Some remarks on the comparability of ideals in semirings

Some remarks on the comparability of ideals in semirings

On $n$-absorbing prime ideals of commutative rings

On 1-Absorbing Fuzzy Ideals of Commutative Semirings

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