More on the annihilator-ideal graph of a commutative ring

Let [Formula: see text] be a commutative ring with identity and [Formula: see text] be the set of ideals of [Formula: see text] with nonzero annihilator. The annihilator-ideal graph of [Formula: see text], denoted by [Formula: see text], is a simple graph with the vertex set [Formula: see text], and two distinct vertices [Formula: see text] and [Formula: see text] are adjacent if and only if [Formula: see text]. In this paper, we study the affinity between the annihilator-ideal graph and the annihilating-ideal graph [Formula: see text] (a well known graph with the same vertices and two distinct vertices [Formula: see text] are adjacent if and only if [Formula: see text]) associated with [Formula: see text]. All rings whose [Formula: see text] and [Formula: see text] are characterized. Among other results, we obtain necessary and sufficient conditions under which [Formula: see text] is a star graph.

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Inertial subalgebras of algebras possessing finite automorphism groups

Journal of the Australian Mathematical Society ◽

10.1017/s1446788700012295 ◽

1979 ◽

Vol 28 (3) ◽

pp. 335-345 ◽

Cited By ~ 1

Author(s):

Nicholas S. Ford

Keyword(s):

Finite Group ◽

Commutative Ring ◽

Jacobson Radical ◽

Automorphism Groups ◽

Sufficient Conditions ◽

Necessary And Sufficient Conditions ◽

Finitely Generated ◽

Fixed Ring ◽

A Finite Group ◽

Necessary And Sufficient

AbstractLet R be a commutative ring with identity, and let A be a finitely generated R-algebra with Jacobson radical N and center C. An R-inertial subalgebra of A is a R-separable subalgebra B with the property that B+N=A. Suppose A is separable over C and possesses a finite group G of R-automorphisms whose restriction to C is faithful with fixed ring R. If R is an inertial subalgebra of C, necessary and sufficient conditions for the existence of an R-inertial subalgebra of A are found when the order of G is a unit in R. Under these conditions, an R-inertial subalgebra B of A is characterized as being the fixed subring of a group of R-automorphisms of A. Moreover, A ⋍ B ⊗R C. Analogous results are obtained when C has an R-inertial subalgebra S ⊃ R.

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When the annihilator-ideal graph is planar or complete bipartite graph

Asian-European Journal of Mathematics ◽

10.1142/s1793557119500244 ◽

2019 ◽

Vol 12 (02) ◽

pp. 1950024

Author(s):

M. J. Nikmehr ◽

S. M. Hosseini

Keyword(s):

Commutative Ring ◽

Bipartite Graph ◽

Complete Bipartite Graph ◽

Simple Graph ◽

Annihilator Ideal ◽

Vertex Set

Let [Formula: see text] be a commutative ring with identity and [Formula: see text] be the set of ideals of [Formula: see text] with nonzero annihilator. The annihilator-ideal graph of [Formula: see text], denoted by [Formula: see text], is a simple graph with the vertex set [Formula: see text], and two distinct vertices [Formula: see text] and [Formula: see text] are adjacent if and only if [Formula: see text]. In this paper, we present some results on the bipartite, complete bipartite, outer planar and unicyclic of the annihilator-ideal graphs of a commutative ring. Among other results, bipartite annihilator-ideal graphs of rings are characterized. Also, we investigate planarity of the annihilator-ideal graph and classify rings whose annihilator-ideal graph is planar.

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Some results on the complement of the annihilating ideal graph of a commutative ring

Journal of Algebra and Its Applications ◽

10.1142/s0219498815500991 ◽

2015 ◽

Vol 14 (07) ◽

pp. 1550099 ◽

Cited By ~ 3

Author(s):

S. Visweswaran ◽

Hiren D. Patel

Keyword(s):

Commutative Ring ◽

Sufficient Conditions ◽

Necessary And Sufficient Conditions ◽

Sufficient Condition ◽

Necessary And Sufficient Condition ◽

Necessary And Sufficient

Rings considered in this article are commutative with identity which admit at least one nonzero annihilating ideal. For such a ring R, we determine necessary and sufficient conditions in order that the complement of its annihilating ideal graph is connected and also find its diameter when it is connected. We discuss the girth of the complement of the annihilating ideal graph of R and prove that it is either equal to 3 or ∞. We also present a necessary and sufficient condition for the complement of the annihilating ideal graph to be complemented.

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COMMUTATIVITY OF SKEW SYMMETRIC ELEMENTS IN GROUP RINGS

Proceedings of the Edinburgh Mathematical Society ◽

10.1017/s0013091504000896 ◽

2007 ◽

Vol 50 (1) ◽

pp. 37-47 ◽

Cited By ~ 8

Author(s):

Osnel Broche Cristo ◽

César Polcino Milies

Keyword(s):

Commutative Ring ◽

Group Ring ◽

Sufficient Conditions ◽

Necessary And Sufficient Conditions ◽

Group Rings ◽

Necessary And Sufficient

AbstractLet $R$ be a commutative ring with unity and let $G$ be a group. The group ring $RG$ has a natural involution that maps each element $g\in G$ to its inverse. We denote by $RG^-$ the set of skew symmetric elements under this involution. We study necessary and sufficient conditions for $RG^-$ to be commutative.

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Pairs of rings with a bijective correspondence between the prime spectra

Journal of the Australian Mathematical Society. Series A. Pure Mathematics and Statistics ◽

10.1017/s1446788700032055 ◽

1993 ◽

Vol 55 (2) ◽

pp. 238-245

Author(s):

E. Jespers ◽

P. Wauters

Keyword(s):

Abelian Group ◽

Commutative Ring ◽

Free Abelian Group ◽

Sufficient Conditions ◽

Necessary And Sufficient Conditions ◽

Polynomial Rings ◽

Prime Spectrum ◽

Bijective Correspondence ◽

Natural Mapping ◽

Necessary And Sufficient

AbstractLet A be a subring of a commutative ring B. If the natural mapping from the prime spectrum of B to the prime spectrum of A is injective (respectively bijective) then the pair (A, B) is said to have the injective (respectively bijective) Spec-map. We give necessary and sufficient conditions for a pair of rings A and B graded by a free abelian group to have the injective (respectively bijective) Spec-map. For this we first deal with the polynomial case. Let l be a field and k a subfield. Then the pair of polynomial rings (k[X], l[X]) has the injective Spec-map if and only if l is a purely inseparable extension of k.

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COMMUTATIVE TALL RINGS

Journal of Algebra and Its Applications ◽

10.1142/s0219498813501296 ◽

2014 ◽

Vol 13 (04) ◽

pp. 1350129

Author(s):

TOMÁŠ PENK ◽

JAN ŽEMLIČKA

Keyword(s):

Commutative Ring ◽

Sufficient Conditions ◽

Commutative Rings ◽

Necessary And Sufficient Conditions ◽

Theoretical Criterion ◽

Necessary And Sufficient

A ring is right tall if every non-noetherian right module contains a proper non-noetherian submodule. We prove a ring-theoretical criterion of tall commutative rings. Besides other examples which illustrate limits of proven necessary and sufficient conditions, we construct an example of a tall commutative ring that is non-max.

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Extending abelian groups to rings

Journal of the Australian Mathematical Society ◽

10.1017/s1446788700036132 ◽

2007 ◽

Vol 82 (3) ◽

pp. 297-314 ◽

Cited By ~ 2

Author(s):

Lynn M. Batten ◽

Robert S. Coulter ◽

Marie Henderson

Keyword(s):

Abelian Group ◽

Commutative Ring ◽

Equivalence Relation ◽

Binary Operation ◽

Sufficient Conditions ◽

Additive Group ◽

Necessary And Sufficient Conditions ◽

Odd Order ◽

Weak Isomorphism ◽

Necessary And Sufficient

AbstractFor any abelian group G and any function f: G → G we define a commutative binary operation or ‘multiplication’ on G in terms of f. We give necessary and sufficient conditions on f for G to extend to a commutative ring with the new multiplication. In the case where G is an elementary abelian p–group of odd order, we classify those functions which extend G to a ring and show, under an equivalence relation we call weak isomorphism, that there are precisely six distinct classes of rings constructed using this method with additive group the elementary abelian p–group of odd order p2.

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Some Extensions of Generalized Morphic Rings and EM-rings

Analele Universitatii Ovidius Constanta - Seria Matematica ◽

10.2478/auom-2018-0007 ◽

2018 ◽

Vol 26 (1) ◽

pp. 111-123

Author(s):

Manal Ghanem ◽

Emad Abu Osba

Keyword(s):

Commutative Ring ◽

Sufficient Conditions ◽

Necessary And Sufficient Conditions ◽

Polynomial Rings ◽

Necessary And Sufficient

AbstractLet R be a commutative ring with unity. The main objective of this article is to study the relationships between PP-rings, generalized morphic rings and EM-rings. Although PP-rings are included in the later rings, the converse is not in general true. We put necessary and sufficient conditions to ensure the converse using idealization and polynomial rings

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Restricted triangulation on circulant graphs

Open Mathematics ◽

10.1515/math-2018-0033 ◽

2018 ◽

Vol 16 (1) ◽

pp. 358-369 ◽

Cited By ~ 1

Author(s):

Niran Abbas Ali ◽

Adem Kilicman ◽

Hazim Michman Trao

Keyword(s):

Sufficient Conditions ◽

Necessary And Sufficient Conditions ◽

Existence Problem ◽

Circulant Graph ◽

Circulant Graphs ◽

Value Set ◽

Vertex Set ◽

Necessary And Sufficient

AbstractThe restricted triangulation existence problem on a given graph decides whether there exists a triangulation on the graph’s vertex set that is restricted with respect to its edge set. Let G = C(n, S) be a circulant graph on n vertices with jump value set S. We consider the restricted triangulation existence problem for G. We determine necessary and sufficient conditions on S for which G admitting a restricted triangulation. We characterize a set of jump values S(n) that has the smallest cardinality with C(n, S(n)) admits a restricted triangulation. We present the measure of non-triangulability of Kn − G for a given G.

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On the Number of Quasi-Kernels in Digraphs

BRICS Report Series ◽

10.7146/brics.v8i7.20461 ◽

2001 ◽

Vol 8 (7) ◽

Author(s):

Gregory Gutin ◽

Khee Meng Koh ◽

Eng Guan Tay ◽

Anders Yeo

Keyword(s):

Sufficient Conditions ◽

Necessary And Sufficient Conditions ◽

Vertex Set ◽

Necessary And Sufficient

A vertex set X of a digraph D = (V,A) is a kernel if X is independent (i.e., all pairs of distinct vertices of X are non-adjacent) and for every v in V − X there exists x in X such that vx in A. A vertex set X of a digraph D = (V,A) is a quasi-kernel if X is independent and for every v in V − X there exist w in V − X, x in X such that either vx in A or vw,wx in A: In 1994, Chvatal and Lovasz proved that every digraph has a quasi-kernel. In 1996, Jacob and Meyniel proved that, if a digraph D has no kernel, then D contains at least three quasi-kernels.We characterize digraphs with exactly one and two quasi-kernels, and, thus, provide necessary and sufficient conditions for a digraph to have at least three quasi-kernels. In particular, we prove that every strong digraph of order at least three, which is not a 4-cycle, has at least three quasi-kernels. We conjecture that every digraph with no sink has a pair of disjoint quasi-kernels and provide some support to this conjecture.

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