scholarly journals When Is the Complement of the Comaximal Graph of a Commutative Ring Planar?

ISRN Algebra ◽  
2014 ◽  
Vol 2014 ◽  
pp. 1-8 ◽  
Author(s):  
S. Visweswaran ◽  
Jaydeep Parejiya
Keyword(s):  
Commutative Ring ◽  
Jacobson Radical ◽  
Comaximal Graph

Let R be a commutative ring with identity. In this paper we classify rings R such that the complement of comaximal graph of R is planar. We also consider the subgraph of the complement of comaximal graph of R induced on the set S of all nonunits of R with the property that each element of S is not in the Jacobson radical of R and classify rings R such that this subgraph is planar.

On the Comaximal Graph of a Commutative Ring

2014 ◽  
Vol 57 (2) ◽  
pp. 413-423 ◽  
Author(s):  
Karim Samei
Keyword(s):  
Commutative Ring ◽  
Jacobson Radical ◽  
Topological Properties ◽  
Dominating Sets ◽  
Graph Properties ◽  
Comaximal Graph

AbstractLet R be a commutative ring with 1. In a 1995 paper in J. Algebra, Sharma and Bhatwadekar defined a graph on R, Γ(R), with vertices as elements of R, where two distinct vertices a and b are adjacent if and only if Ra + Rb = R. In this paper, we consider a subgraph Γ2(R) of Γ(R) that consists of non-unit elements. We investigate the behavior of Γ2(R) and Γ2(R)\J(R), where J(R) is the Jacobson radical of R. We associate the ring properties of R, the graph properties of Γ(R), and the topological properties of Max(R). Diameter, girth, cycles and dominating sets are investigated, and algebraic and topological characterizations are given for graphical properties of these graphs.

scholarly journals Inertial subalgebras of algebras possessing finite automorphism groups

1979 ◽  
Vol 28 (3) ◽  
pp. 335-345 ◽  
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.

scholarly journals Complementation of the Jacobson group in a matrix ring

2005 ◽  
Vol 72 (2) ◽  
pp. 317-324
Author(s):  
David Dolžan
Keyword(s):  
Normal Subgroup ◽  
Commutative Ring ◽  
Jacobson Radical ◽  
Matrix Ring ◽  
Graded Rings ◽  
Matrix Rings ◽  
Generators And Relations ◽  
Group Of Units ◽  
The Matrix ◽  
Finite Commutative Ring

The Jacobson group of a ring R (denoted by  = (R)) is the normal subgroup of the group of units of R (denoted by G(R)) obtained by adding 1 to the Jacobson radical of R (J(R)). Coleman and Easdown in 2000 showed that the Jacobson group is complemented in the group of units of any finite commutative ring and also in the group of units a n × n matrix ring over integers modulo ps, when n = 2 and p = 2, 3, but it is not complemented when p ≥ 5. In 2004 Wilcox showed that the answer is positive also for n = 3 and p = 2, and negative in all the remaining cases. In this paper we offer a different proof for Wilcox's results and also generalise the results to a matrix ring over an arbitrary finite commutative ring. We show this by studying the generators and relations that define a matrix ring over a field. We then proceed to examine the complementation of the Jacobson group in the matrix rings over graded rings and prove that complementation depends only on the 0-th grade.

When is the Jacobson graph projective?

2018 ◽  
Vol 17 (09) ◽  
pp. 1850168
Author(s):  
Atossa Parsapour ◽  
Khadijeh Ahmadjavaheri
Keyword(s):  
Commutative Ring ◽  
Jacobson Radical ◽  
Jacobson Graph ◽  
Vertex Set ◽  
Finite Commutative Ring ◽  
Nonzero Identity

Let [Formula: see text] be a commutative ring with nonzero identity and [Formula: see text] be the Jacobson radical of [Formula: see text]. The Jacobson graph of [Formula: see text], denoted by [Formula: see text], is a graph with vertex-set [Formula: see text], such that two distinct vertices [Formula: see text] and [Formula: see text] in [Formula: see text] are adjacent if and only if [Formula: see text] is not a unit of [Formula: see text]. The goal in this paper is to list every finite commutative ring [Formula: see text] with nonzero identity (up to isomorphism) such that the graph [Formula: see text] is projective.

On the Jacobson radical of certain commutative semigroup algebras

1984 ◽  
Vol 96 (1) ◽  
pp. 15-23 ◽  
Author(s):  
W. D. Munn
Keyword(s):  
Commutative Ring ◽  
Commutative Semigroup ◽  
Jacobson Radical ◽  
Semigroup Algebras

In two previous papers the author studied the Jacobson and nil redicals of the algebra of a commutative semigroup over a field [8] and over a commutative ring with unity [9]. This work is continued here.

CO-MAXIMAL IDEAL GRAPHS OF COMMUTATIVE RINGS

2012 ◽  
Vol 11 (06) ◽  
pp. 1250114 ◽  
Author(s):  
MENG YE ◽  
TONGSUO WU
Keyword(s):  
Commutative Ring ◽  
Chromatic Number ◽  
Maximal Ideal ◽  
Connected Graph ◽  
Jacobson Radical ◽  
Clique Number ◽  
Commutative Rings ◽  
Maximal Ideals ◽  
Simple Connected Graph

In this paper, a new kind of graph on a commutative ring R with identity, namely the co-maximal ideal graph is defined and studied. We use [Formula: see text] to denote this graph, with its vertices the proper ideals of R which are not contained in the Jacobson radical of R, and two vertices I1 and I2 are adjacent if and only if I1 + I2 = R. We show some properties of this graph. For example, this graph is a simple, connected graph with diameter less than or equal to three, and both the clique number and the chromatic number of the graph are equal to the number of maximal ideals of the ring R.

On Group Rings

1970 ◽  
Vol 22 (2) ◽  
pp. 249-254 ◽  
Author(s):  
D. B. Coleman
Keyword(s):  
Nilpotent Group ◽  
Commutative Ring ◽  
Group Ring ◽  
Characteristic Zero ◽  
Sylow Subgroup ◽  
Jacobson Radical ◽  
Closed Field ◽  
Group Rings ◽  
Locally Finite ◽  
Image Position

Let R be a commutative ring with unity and let G be a group. The group ring RG is a free R-module having the elements of G as a basis, with multiplication induced byThe first theorem in this paper deals with idempotents in RG and improves a result of Connell. In the second section we consider the Jacobson radical of RG, and we prove a theorem about a class of algebras that includes RG when G is locally finite and R is an algebraically closed field of characteristic zero. The last theorem shows that if R is a field and G is a finite nilpotent group, then RG determines RP for every Sylow subgroup P of G, regardless of the characteristic of R.

The Jacobson Radical and Regular Modules

1975 ◽  
Vol 18 (1) ◽  
pp. 141-141
Author(s):  
David J. Fieldhouse
Keyword(s):  
Commutative Ring ◽  
Jacobson Radical

Let A be an associative, but not necessarily commutative, ring with identity, and J = J(A) its Jacobson radical. A (unital) module is regular iff every submodule is pure (see (1)). The regular socle R(M) of a module M is the sum of all its submodules which are regular. These concepts have been introduced and studied in (2).

The genus two class of graphs arising from rings

2018 ◽  
Vol 17 (10) ◽  
pp. 1850193 ◽  
Author(s):  
T. Asir
Keyword(s):  
Commutative Ring ◽  
Jacobson Radical ◽  
Commutative Rings ◽  
Isomorphism Classes ◽  
Jacobson Graph ◽  
Vertex Set ◽  
Genus Two

Given a commutative ring [Formula: see text] with identity [Formula: see text], its Jacobson graph [Formula: see text] is defined to be the graph in which the vertex set is [Formula: see text], and two distinct vertices [Formula: see text] and [Formula: see text] are adjacent if and only if [Formula: see text]. Here [Formula: see text] denotes the Jacobson radical of [Formula: see text] and [Formula: see text] is the set of unit elements in [Formula: see text]. This paper investigates the genus properties of Jacobson graph. In particular, we determine all isomorphism classes of commutative rings whose Jacobson graph has genus two.

Kurosh Radicals of Rings with Operators

1965 ◽  
Vol 17 ◽  
pp. 278-280 ◽  
Author(s):  
N. Divinsky ◽  
A. Suliński
Keyword(s):  
Commutative Ring ◽  
Jacobson Radical ◽  
Present Note

If A is an algebra over a commutative ring with unity, Φ, then the Jacobson radical of the algebra A is equal to the Jacobson radical of A, thought of as a ring (1, p. 18, Theorem 1). The present note extends this result to all radical properties (in the sense of Kurosh 2) and allows ϕ to be any set of operators on A.If A is a ring and Φ is an arbitrary set, we say that Φ is a set of operators for A if for any α in Φ and any x in A, the composition ax is defined and is an element of A, and if this composition satisfies the following two conditions:

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