Graded rings and essential ideals

Hongjin, Fang; Stewart, Patrick N.

doi:10.1007/bf02560128

Cited by 3 publications

(2 citation statements)

References 6 publications

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“…Moreover, essential ideals have been investigated in rings of measurable functions [13] and C * -algebras [10]. For more on essential ideals, see [3,8,9,20].…”

Section: Introductionmentioning

confidence: 99%

Essential ideals represented by mod-annihilators of modules

Raja¹,

Pirzada²

2022

Preprint

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Let R be a commutative ring with unity, M be a unitary R-module and G a finite abelian group (viewed as a Z-module). The main objective of this paper is to study properties of mod-annihilators of M. For x ∈ M, we study the ideals [x : M] = {r ∈ R | rM ⊆ Rx} of R corresponding to mod-annihilator of M. We investigate that when [x : M] is an essential ideal of R. We prove that arbitrary intersection of essential ideals represented by mod-annihilators is an essential ideal. We observe that [x : M] is injective if and only if R is non-singular and the radical of R/[x : M] is zero. Moreover, if essential socle of M is non-zero, then we show that [x : M] is the intersection of maximal ideals and [x : M] 2 = [x : M]. Finally, we discuss the correspondence of essential ideals of R and vertices of the annihilating graphs realized by M over R.

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“…Moreover, essential ideals have been investigated in rings of measurable functions [13] and C * -algebras [10]. For more on essential ideals, see [3,8,9,20].…”

Section: Introductionmentioning

confidence: 99%

Essential ideals represented by mod-annihilators of modules

Raja¹,

Pirzada²

2022

Preprint

View full text Add to dashboard Cite

show abstract

“…Moreover, essential ideals also have been investigated in rings of measurable functions [18] and C * -algebras [14]. For more on essential ideals see [12,13,22].…”

Section: Introductionmentioning

confidence: 99%

On combinatorial aspects of modules over commutative rings

Raja

2017

Preprint

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Let R be a commutative ring with unity, M be an unitary Rmodule and Γ be a simple graph. This research article is an interplay of combinatorial and algebraic properties of M . We show a combinatorial object completely determines an algebraic object and characterize all finite abelian groups. We discuss the correspondence between essential ideals of R, submodules of M and vertices of graphs arising from M . We examine various types of equivalence relations on objects A f (M ), As(M ) and At(M ), where A f (M ) is an object of full-annihilators, As(M ) is an object of semi-annihilators and At(M ) is an object of star-annihilators in M . We study essential ideals corresponding to elements of an object A f (M ) over hereditary and regular rings. Further, we study isomorphism of annihilating graphs arising from M and tensor product M ⊗ R T −1 R, where T = R\C(M ), where C(M ) = {r ∈ R : rm = 0 f or some 0 = m ∈ M }, and show that ann f (Γ(M ⊗ R T −1 R)) ∼ = ann f (Γ(M )) for every R-module M .

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Essential ideals represented by mod-annihilators of modules

Raja

Pirzada

2022

Afr. Mat.

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Graded rings and essential ideals

Cited by 3 publications

References 6 publications

Essential ideals represented by mod-annihilators of modules

Essential ideals represented by mod-annihilators of modules

On combinatorial aspects of modules over commutative rings

Essential ideals represented by mod-annihilators of modules

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