On Weakly δ-Semiprimary Ideals of Commutative Rings (original) (raw)
Related papers
On weakly (m, n)−closed δ−primary ideals of commutative rings
Let R be a commutative ring with 1 ̸ = 0. In this article, we introduce the concept of weakly (m, n)−closed δ−primary ideals of R and explore its basic properties. We show that a proper ideal I of R is a weakly (m, n)−closed γ • δ−primary ideal of R if and only if I is an (m, n)−closed γ • δ−primary ideal of R, where δ and γ are expansions ideals of R with δ(0) is an (m, n)−closed γ−primary ideal of R. Furthermore, we provide examples to demonstrate the validity and applicability of our results.
On weakly 2-absorbing δ-primary ideals of commutative rings
Georgian Mathematical Journal, 2018
Let R be a commutative ring with {1\neq 0}. We recall that 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\not=abc\in I}, then {ab\in I} or {ac\in\sqrt{I}} or {bc\in\sqrt{I}}. In this paper, we introduce a new class of ideals that is closely related to the class of weakly 2-absorbing primary ideals. Let {I(R)} be the set of all ideals of R and let {\delta:I(R)\rightarrow I(R)} be a function. Then δ is called an expansion function of ideals of R if whenever {L,I,J} are ideals of R with {J\subseteq I}, then {L\subseteq\delta(L)} and {\delta(J)\subseteq\delta(I)}. Let δ be an expansion function of ideals of R. Then a proper ideal I of R (i.e., {I\not=R}) is called a weakly 2-absorbing δ-primary ideal if {0\not=abc\in I} implies {ab\in I} or {ac\in\delta(I)} or {bc\in\delta(I)}. For example, let {\delta:I(R)\rightarrow I(R)} such that {\delta(I)=\sqrt{I}}. Then δ is an expansion function of ideals of R, and hence a proper ideal I o...
On weakly semiprime ideals of commutative rings
Beiträge zur Algebra und Geometrie / Contributions to Algebra and Geometry, 2016
Let R be a commutative ring with identity 1 = 0 and let I be a proper ideal of R. D. D. Anderson and E. Smith called I weakly prime if a, b ∈ R and 0 = ab ∈ I implies a ∈ I or b ∈ I. In this paper, we define I to be weakly semiprime if a ∈ R and 0 = a 2 ∈ I implies a ∈ I. For example, every proper ideal of a quasilocal ring (R, M) with M 2 = 0 is weakly semiprime. We give examples of weakly semiprime ideals that are neither semiprime nor weakly prime. We show that a weakly semiprime ideal of R that is not semiprime is a nil ideal of R. We show that if I is a weakly semiprime ideal of R that is not semiprime and 2 is not a zero-divisor of of R, then I 2 = {0} (and hence i 2 = 0 for every i ∈ I). We give an example of a ring R that admits a weakly semiprime ideal I that is not semiprime where i 2 = 0 for some i ∈ I. If R = R 1 × R 2 for some rings R 1 , R 2 , then we characterize all weakly semiprime ideals of R that are not semiprime. We characterize all weakly semiprime ideals of of Z m that are not semiprime. We show that every proper ideal of R is weakly semiprime if and only if either R is von Neumann regular or R is quasilocal with maximal ideal Nil(R) such that w 2 = 0 for every w ∈ Nil(R). Keywords Primary ideal • Prime ideal • Weakly prime ideal • 2-absorbing ideal • n-absorbing ideal • Semiprime • Weakly semiprime ideal
2016
Let R be a commutative ring with identity 1 = 0 and let I be a proper ideal of R. D. D. Anderson and E. Smith called I weakly prime if a, b ∈ R and 0 = ab ∈ I implies a ∈ I or b ∈ I. In this paper, we define I to be weakly semiprime if a ∈ R and 0 = a 2 ∈ I implies a ∈ I. For example, every proper ideal of a quasilocal ring (R, M) with M 2 = 0 is weakly semiprime. We give examples of weakly semiprime ideals that are neither semiprime nor weakly prime. We show that a weakly semiprime ideal of R that is not semiprime is a nil ideal of R. We show that if I is a weakly semiprime ideal of R that is not semiprime and 2 is not a zero-divisor of of R, then I 2 = {0} (and hence i 2 = 0 for every i ∈ I). We give an example of a ring R that admits a weakly semiprime ideal I that is not semiprime where i 2 = 0 for some i ∈ I. If R = R 1 × R 2 for some rings R 1 , R 2 , then we characterize all weakly semiprime ideals of R that are not semiprime. We characterize all weakly semiprime ideals of of Z m that are not semiprime. We show that every proper ideal of R is weakly semiprime if and only if either R is von Neumann regular or R is quasilocal with maximal ideal Nil(R) such that w 2 = 0 for every w ∈ Nil(R). Keywords Primary ideal · Prime ideal · Weakly prime ideal · 2-absorbing ideal · n-absorbing ideal · Semiprime · Weakly semiprime ideal Mathematics Subject Classification Primary 13A15; Secondary 13F05 · 13G05
On Weakly Semiprime Ideals in Commutative Rings
2021
In this paper, we study weakly semiprime ideals of a commutative ring with nonzeroidentity. We give some properties of such ideals. Also, we investigate some results of weakly semiprime submodules of a module over a commutative ring RRR with nonzero identity.
On 2-Absorbing and Weakly 2-Absorbing Ideals of Commutative Semirings
Kyungpook mathematical journal, 2012
Let R be a commutative semiring. We define a proper ideal I of R to be 2-absorbing (resp., weakly 2-absorbing) if abc ∈ I (resp., 0 ̸ = abc ∈ I) implies ab ∈ I or ac ∈ I or bc ∈ I. We show that a weakly 2-absorbing ideal I with I 3 ̸ = 0 is 2-absorbing. We give a number of results concerning 2-absorbing and weakly 2-absorbing ideals and examples of weakly 2-absorbing ideals. Finally we define the concept of 0 − (1−, 2−, 3−)2absorbing ideals of R and study the relationship among these classes of ideals of R.
On weakly S-prime ideals of commutative rings
Analele Universitatii "Ovidius" Constanta - Seria Matematica, 2021
Let R be a commutative ring with identity and S be a multiplicative subset of R. In this paper, we introduce the concept of weakly S-prime ideals which is a generalization of weakly prime ideals. Let P be an ideal of R disjoint with S. We say that P is a weakly S-prime ideal of R if there exists an s ∈ S such that, for all a, b ∈ R, if 0 ≠ ab ∈ P, then sa ∈ P or sb ∈ P. We show that weakly S-prime ideals have many analog properties to these of weakly prime ideals. We also use this new class of ideals to characterize S-Noetherian rings and S-principal ideal rings.
ON WEAKLY 2-ABSORBING IDEALS of Commutative Rings
atlas-conferences.com
Let R be a commutative ring with identity . Various generalizations of prime ideals have been studied. For example, a proper ideal I of R is weakly prime if a, b in R with ab in I and ab is nonzero , then either a in I or b in I. Also a proper ideal I of R is said to be 2-absorbing if whenever ...