• Communications of the Korean Mathematical Society
• /
• v.13 no.2
• /
• pp.251-256
• /
• 1998

We introduce the concept of weakly prime ideals in po-$\Gamma$-semigroup and give some characterizations of weakly prime ideals.

• Kyungpook Mathematical Journal
• /
• v.56 no.4
• /
• pp.1085-1101
• /
• 2016

In this paper, all rings are commutative with nonzero identity. Let M be an R-module. A proper submodule N of M is called a classical prime submodule, if for each $m{\in}M$ and elements a, $b{\in}R$, $abm{\in}N$ implies that $am{\in}N$ or $bm{\in}N$. We introduce the concept of "weakly classical prime submodules" and we will show that this class of submodules enjoys many properties of weakly 2-absorbing ideals of commutative rings. A proper submodule N of M is a weakly classical prime submodule if whenever $a,b{\in}R$ and $m{\in}M$ with $0{\neq}abm{\in}N$, then $am{\in}N$ or $bm{\in}N$.

• Bulletin of the Korean Mathematical Society
• /
• v.47 no.5
• /
• pp.1077-1087
• /
• 2010

Anderson-Smith [1] studied weakly prime ideals for a commutative ring with identity. Blair-Tsutsui [2] studied the structure of a ring in which every ideal is prime. In this paper we investigate the structure of rings, not necessarily commutative, in which all ideals are weakly prime.

• The Pure and Applied Mathematics
• /
• v.12 no.3 s.29
• /
• pp.193-209
• /
• 2005

We introduce the notions of intuitionistic fuzzy prime ideals, intuitionistic fuzzy completely prime ideals and intuitionistic fuzzy weakly completely prime ideals. And we give a characterization of intuitionistic fuzzy ideals and establish relationships between intuitionistic fuzzy completely prime ideals and intuitionistic fuzzy weakly completely prime ideals.

• Kyungpook Mathematical Journal
• /
• v.47 no.4
• /
• pp.473-480
• /
• 2007

In this paper we define prime right ideals of semirings and prove that if every right ideal of a semiring R is prime then R is weakly regular. We also prove that if the set of right ideals of R is totally ordered then every right ideal of R is prime if and only if R is right weakly regular. Moreover in this paper we also define prime subsemimodule (generalizing the concept of prime right ideals) of an R-semimodule. We prove that if a subsemimodule K of an R-semimodule M is prime then $A_K(M)$ is also a prime ideal of R.

• International Journal of Fuzzy Logic and Intelligent Systems
• /
• v.12 no.3
• /
• pp.198-204
• /
• 2012

We introduce the notions of interval-valued fuzzy prime ideals, interval-valued fuzzy completely prime ideals and interval-valued fuzzy weakly completely prime ideals. And we give a characterization of interval-valued fuzzy ideals and establish relationships between interval-valued fuzzy completely prime ideals and interval-valued fuzzy weakly completely prime ideals.

• Bulletin of the Korean Mathematical Society
• /
• v.44 no.1
• /
• pp.151-156
• /
• 2007

A connection between weak ${\pi}-regularity$ and the condition every prime ideal is maximal will be investigated. We prove that a certain 2-primal ring R is weakly ${\pi}-regular$ if and only if every prime ideal is maximal. This result extends several known results nontrivially. Moreover a characterization of minimal prime ideals is also considered.

• Kyungpook Mathematical Journal
• /
• v.54 no.1
• /
• pp.123-129
• /
• 2014

In this paper the concepts of pure ideals, weakly pure ideals and purely prime ideals in ordered semigroups are introduced. We obtain some characterizations of pure ideals and prove that the set of all pure prime ideals is topologized.

• Bulletin of the Korean Mathematical Society
• /
• v.58 no.5
• /
• pp.1069-1078
• /
• 2021

Let R be a commutative ring with identity. A proper ideal I of R is called 1-absorbing primary ([4]) if for all nonunit a, b, c ∈ R such that abc ∈ I, then either ab ∈ I or c ∈ ${\sqrt{1}}$. The concept of 1-absorbing primary ideals in a polynomial ring, in a PID and in idealization of a module is studied. Moreover, we introduce weakly 1-absorbing primary ideals which are generalization of weakly prime ideals and 1-absorbing primary ideals. A proper ideal I of R is called weakly 1-absorbing primary if for all nonunit a, b, c ∈ R such that 0 ≠ abc ∈ I, then either ab ∈ I or c ∈ ${\sqrt{1}}$. Some properties of weakly 1-absorbing primary ideals are investigated. For instance, weakly 1-absorbing primary ideals in decomposable rings are characterized. Among other things, it is proved that if I is a weakly 1-absorbing primary ideal of a ring R and 0 ≠ I₁I₂I₃ ⊆ I for some ideals I₁, I₂, I₃ of R such that I is free triple-zero with respect to I₁I₂I₃, then I₁I₂ ⊆ I or I₃ ⊆ I.

• Journal of the Korean Mathematical Society
• /
• v.52 no.1
• /
• pp.97-111
• /
• 2015

Let R be a commutative ring with $1{\neq}0$. In this paper, we introduce the concept of weakly 2-absorbing primary ideal which is a generalization of weakly 2-absorbing ideal. A proper ideal I of R is called a weakly 2-absorbing primary ideal of R if whenever a, b, $c{\in}R$ and $0{\neq}abc{\in}I$, then $ab{\in}I$ or $ac{\in}\sqrt{I}$ or $bc{\in}\sqrt{I}$. A number of results concerning weakly 2-absorbing primary ideals and examples of weakly 2-absorbing primary ideals are given.