The projective 3-annihilating-ideal hypergraphs

2021 ◽  
pp. 2250005
Author(s):  
V. Ramanathan ◽  
C. Selvaraj
Keyword(s):  
Commutative Ring ◽  
Local Rings ◽  
Topological Embedding ◽  
Projective Graph ◽  
Non Local ◽  
Compact Surfaces

In this paper, we investigate the crosscap of 3-annihilating-ideal hypergraph [Formula: see text] of a commutative ring [Formula: see text] and the topological embedding of [Formula: see text] to the nonorientable compact surfaces. Furthermore, we determine all Artinian commutative non-local rings [Formula: see text] (up to isomorphism) such that [Formula: see text] is a projective graph.

On the genus of the k-maximal hypergraph of commutative rings

2019 ◽  
Vol 11 (01) ◽  
pp. 1950010
Author(s):  
K. Selvakumar ◽  
V. C. Amritha
Keyword(s):  
Commutative Ring ◽  
Commutative Rings ◽  
Local Rings ◽  
Maximal Elements ◽  
Fixed Integer ◽  
Isomorphism Classes ◽  
Vertex Set ◽  
Non Local ◽  
Genus One

Let [Formula: see text] be a commutative ring with identity and [Formula: see text], a fixed integer. Let [Formula: see text] be the set of all [Formula: see text]-maximal elements in [Formula: see text] Associate a [Formula: see text]-maximal hypergraph [Formula: see text] to [Formula: see text] with vertex set [Formula: see text] and for distinct elements [Formula: see text] in [Formula: see text], the set [Formula: see text] is an edge of [Formula: see text] if and only if [Formula: see text] and [Formula: see text] for all [Formula: see text]. In this paper, we determine all isomorphism classes of finite commutative non-local rings with identity whose [Formula: see text]-maximal hypergraph has genus one. Finally, we classify all finite commutative non-local rings [Formula: see text] for which [Formula: see text] is projective.

scholarly journals Correction to: Classification of non-local rings with genus two zero-divisor graphs

2021 ◽  
Vol 25 (4) ◽  
pp. 3355-3356
Author(s):  
T. Asir ◽  
K. Mano ◽  
T. Tamizh Chelvam
Keyword(s):  
Zero Divisor ◽  
Local Rings ◽  
Non Local ◽  
Genus Two

RECOVERING RINGS FROM ZERO-DIVISOR GRAPHS

2013 ◽  
Vol 12 (08) ◽  
pp. 1350047 ◽  
Author(s):  
SHANE P. REDMOND
Keyword(s):  
Commutative Ring ◽  
Local Ring ◽  
Zero Divisor ◽  
Zero Divisor Graph ◽  
Reduced Ring ◽  
Non Local

Suppose G is the zero-divisor graph of some commutative ring with 1. When G has four or more vertices, a method is presented to find a specific commutative ring R with 1 such that Γ(R) ≅ G. Furthermore, this ring R can be written as R ≅ R1 × R2 × ⋯ × Rn, where each Ri is local and this representation of R is unique up to factors Ri with isomorphic zero-divisor graphs. It is also shown that for graphs on four or more vertices, no local ring has the same zero-divisor graph as a non-local ring and no reduced ring has the same zero-divisor graph as a non-reduced ring.

On the genus of the graph associated to a commutative ring

2017 ◽  
Vol 09 (05) ◽  
pp. 1750058 ◽  
Author(s):  
K. Selvakumar ◽  
P. Subbulakshmi ◽  
Jafar Amjadi
Keyword(s):  
Commutative Ring ◽  
Local Ring ◽  
Simple Graph ◽  
Vertex Set ◽  
Non Local ◽  
Genus One

Let [Formula: see text] be a commutative ring with identity. We consider a simple graph associated with [Formula: see text], denoted by [Formula: see text], whose vertex set is the set of all non-trivial ideals of [Formula: see text] and two distinct vertices [Formula: see text] and [Formula: see text] are adjacent whenever [Formula: see text] or [Formula: see text]. In this paper, we characterize the commutative Artinian non-local ring [Formula: see text] for which [Formula: see text] has genus one and crosscap one.

scholarly journals On the Structure of Complete Local Rings

1950 ◽  
Vol 1 ◽  
pp. 63-70 ◽  
Author(s):  
Masayoshi Nagata
Keyword(s):  
Commutative Ring ◽  
Local Ring ◽  
Maximal Ideal ◽  
Noetherian Ring ◽  
Structure Theorem ◽  
Finite Basis ◽  
Local Rings ◽  
Topological Ring

The concept of a local ring was introduced by Krull [2], who defined it as a Noetherian ring R (we say that a commutative ring R is Noetherian if every ideal in R has a finite basis and if R contains the identity) which has only one maximal ideal m. If the powers of m are defined as a system of neighbourhoods of zero, then R becomes a topological ring satisfying the first axiom of countability, And the notion was studied recently by C. Chevalley and I. S. Cohen. Cohen [1] proved the structure theorem for complete rings besides other properties of local rings.

Classification of non-local rings with genus two zero-divisor graphs

2019 ◽  
Vol 24 (1) ◽  
pp. 237-245
Author(s):  
T. Asir ◽  
K. Mano
Keyword(s):  
Zero Divisor ◽  
Local Rings ◽  
Non Local ◽  
Genus Two

scholarly journals Non-local rings whose ideals are all quasi-injective

1972 ◽  
Vol 6 (1) ◽  
pp. 45-52 ◽  
Author(s):  
G. Ivanov
Keyword(s):  
Finite Number ◽  
Local Rings ◽  
Artinian Rings ◽  
Direct Sum ◽  
Non Local ◽  
Left And Right

A ring is a left Q-ring if all of its left ideals are quasi-injective. For an integer m ≤ 2, a sfield D, and a null D-algebra V whose left and right D-dimensions are both equal to one, let H(m, D, V) be the ring of all m x m matrices whose only non-zero entries are arbitrary elements of D along the diagonal and arbitrary elements of V at the places (2, 1), …, (m, m-l) and (l, m). We show that the only indecomposable non-local left Q-rings are the simple artinian rings and the rings H(m, D, V). An arbitrary left Q-ring is the direct sum of a finite number of indecomposable non-local left Q-rings and a Q-ring whose idempotents are all central.

Subrings of the first neighbourhood ring: II

1982 ◽  
Vol 92 (1) ◽  
pp. 35-39
Author(s):  
D. Kirby
Keyword(s):  
Commutative Ring ◽  
Local Ring ◽  
Algebraic Curves ◽  
Present Note ◽  
Quotient Ring ◽  
Singular Points ◽  
Local Rings ◽  
Embedding Dimension ◽  
One Dimensional ◽  
Finite Lattices

In (1) and (2) we studied a lattice of extension rings associated with a commutative ring R with identity. When R, M is a one-dimensional Cohen-Macaulay local ring the elements of are just those integral extensions of R contained in the total quotient ring T(R) and such that lengthR(S/R) is finite. Experiments with local rings of singular points on algebraic curves indicate that only the simplest singularities give rise to finite lattices. So the problem arises as to which local rings R give rise to which finite lattices. In later papers this problem will be investigated in detail, at least when R is of low embedding dimension. The purpose of the present note is to establish some general results which indicate the size of the problem.

Divisor graph of the complement of Γ(R)

2019 ◽  
Vol 12 (04) ◽  
pp. 1950057
Author(s):  
Ravindra Kumar ◽  
Om Prakash
Keyword(s):  
Commutative Ring ◽  
Local Ring ◽  
Integral Domain ◽  
Zero Divisor ◽  
Local Rings ◽  
Zero Divisor Graph ◽  
Finite Commutative Ring

Let [Formula: see text] be the complement of the zero-divisor graph of a finite commutative ring [Formula: see text]. In this paper, we provide the answer of the question (ii) raised by Osba and Alkam in [11] and prove that [Formula: see text] is a divisor graph if [Formula: see text] is a local ring. It is shown that when [Formula: see text] is a product of two local rings, then [Formula: see text] is a divisor graph if one of them is an integral domain. Further, if [Formula: see text], then [Formula: see text] is a divisor graph.

A Note on Quadratic forms Over Arbitrary Semi-Local Rings

1975 ◽  
Vol 27 (3) ◽  
pp. 513-527
Author(s):  
K. I. Mandelberg
Keyword(s):  
Commutative Ring ◽  
Quadratic Forms ◽  
Quadratic Space ◽  
Local Rings ◽  
Quadratic Mapping ◽  
Finitely Generated ◽  
Bilinear Mapping ◽  
Natural Mapping

Let R be a commutative ring. A bilinear space (E, B) over R is â finitely generated projective R-module E together with a symmetric bilinear mapping B:E X E →R which is nondegenerate (i.e. the natural mapping E → HomR(E﹜ R) induced by B is an isomorphism). A quadratic space (E, B, ) is a bilinear space (E, B) together with a quadratic mapping ϕ:E →R such that B(x, y) = ϕ (x + y) — ϕ (x) — ϕ (y) and ϕ (rx) = r2ϕ (x) for all x, y in E and r in R. If 2 is a unit in R, then ϕ (x) = ½. B﹛x,x) and the two types of spaces are in obvious 1 — 1 correspondence.

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