scholarly journals On Polynomial Extensions Of Rings

1956 ◽  
Vol 8 ◽  
pp. 1-2 ◽  
Author(s):  
Michio Yoshida
Keyword(s):  
Commutative Ring ◽  
Unit Element ◽  
Ring Of Polynomials ◽  
Polynomial Extensions

Let A be a commutative ring with unit element, and let A [x] be a ring of polynomials in an indeterminate x with coefficients in A. There are a number of well-known properties which A shares with A [x]. We shall state one of them in the following.

On quasi-radical of near-ring of polynomials

2019 ◽  
Vol 56 (2) ◽  
pp. 252-259
Author(s):  
Ebrahim Hashemi ◽  
Fatemeh Shokuhifar ◽  
Abdollah Alhevaz
Keyword(s):  
Commutative Ring ◽  
Nilpotent Elements ◽  
Ring Of Polynomials

Abstract The intersection of all maximal right ideals of a near-ring N is called the quasi-radical of N. In this paper, first we show that the quasi-radical of the zero-symmetric near-ring of polynomials R0[x] equals to the set of all nilpotent elements of R0[x], when R is a commutative ring with Nil (R)2 = 0. Then we show that the quasi-radical of R0[x] is a subset of the intersection of all maximal left ideals of R0[x]. Also, we give an example to show that for some commutative ring R the quasi-radical of R0[x] coincides with the intersection of all maximal left ideals of R0[x]. Moreover, we prove that the quasi-radical of R0[x] is the greatest quasi-regular (right) ideal of it.

Completely Indecomposable Modules

1949 ◽  
Vol 1 (2) ◽  
pp. 125-152 ◽  
Author(s):  
Ernst Snapper
Keyword(s):  
Abelian Group ◽  
Commutative Ring ◽  
Chain Condition ◽  
Indecomposable Module ◽  
Unit Element ◽  
Indecomposable Modules ◽  
Operator Domain ◽  
Unit Operator ◽  
Ascending Chain Condition

The purpose of this paper is to investigate completely indecomposable modules. A completely indecomposable module is an additive abelian group with a ring A as operator domain, where the following four conditions are satisfied.1-1. A is a commutative ring and has a unit element which is unit operator for .1-2. The submodules of satisfy the ascending chain condition. (Submodule will always mean invariant submodule.)

Generalized unit and unitary Cayley graphs of finite rings

2019 ◽  
Vol 18 (01) ◽  
pp. 1950006 ◽  
Author(s):  
T. Tamizh Chelvam ◽  
S. Anukumar Kathirvel
Keyword(s):  
Commutative Ring ◽  
Undirected Graph ◽  
Unit Element ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Finite Rings ◽  
Vertex Set ◽  
Finite Commutative Ring ◽  
Necessary And Sufficient ◽  
Nonzero Identity

Let [Formula: see text] be a finite commutative ring with nonzero identity and [Formula: see text] be the set of all units of [Formula: see text] The graph [Formula: see text] is the simple undirected graph with vertex set [Formula: see text] in which two distinct vertices [Formula: see text] and [Formula: see text] are adjacent if and only if there exists a unit element [Formula: see text] in [Formula: see text] such that [Formula: see text] is a unit in [Formula: see text] In this paper, we obtain degree of all vertices in [Formula: see text] and in turn provide a necessary and sufficient condition for [Formula: see text] to be Eulerian. Also, we give a necessary and sufficient condition for the complement [Formula: see text] to be Eulerian, Hamiltonian and planar.

Local multiplication maps on F[x]

2014 ◽  
Vol 14 (03) ◽  
pp. 1550029
Author(s):  
Kelly Aceves ◽  
Manfred Dugas
Keyword(s):  
Commutative Ring ◽  
Finite Fields ◽  
Linear Transformation ◽  
Ring Of Polynomials

Let F be a field and A a F-algebra. The F-linear transformation φ : A → A is a local multiplication map if for all a ∈ A there exists some ua ∈ A such that φ(a) = aua. Let [Formula: see text] denote the F-algebra of all local multiplication maps of A. If F is infinite and F[x] is the ring of polynomials over F, then it is known Lemma 1 in [J. Buckner and M. Dugas, Quasi-Localizations of ℤ, Israel J. Math.160 (2007) 349–370] that [Formula: see text]. The purpose of this paper is to study [Formula: see text] for finite fields F. It turns out that in this case [Formula: see text] is a "very" non-commutative ring of cardinality 2ℵ0 with many interesting properties.

scholarly journals A Remark on the Commutativity of Algebraic Rings

1959 ◽  
Vol 14 ◽  
pp. 39-44 ◽  
Author(s):  
Tadasi Nakayama
Keyword(s):  
Commutative Ring ◽  
Unit Element ◽  
Algebraic Ring

Let F be a commutative ring with unit element 1. A ring R is called an algebraic ring over F by Drazin, in his recent paper [2], when R is an algebra over F and every element x of R satisfies an equation of lower monic form

scholarly journals Note on a paper of Tsuzuku

1964 ◽  
Vol 6 (4) ◽  
pp. 196-197
Author(s):  
H. K. Farahat
Keyword(s):  
Symmetric Group ◽  
Commutative Ring ◽  
Permutation Group ◽  
Matrix Representation ◽  
Invertible Element ◽  
Unit Element ◽  
Transitive Permutation Group ◽  
Natural Representation ◽  
Correct Proof ◽  
Permutation Matrices

In [2], Tosiro Tsuzzuku gave a proof of the following:THEOREM. Let G be a doubly transitive permutation group of degree n, let K be any commutative ring with unit element and let p be the natural representation of G by n × n permutation matrices with elements 0, 1 in K. Then ρ is decomposable as a matrix representation over K if and only ifn is an invertible element of K.For G the symmetric group this result follows from Theorems (2.1) and (4.12) of [1]. The proof given by Tsuzuku is unsatisfactory, although it is perfectly valid when K is a field. The purpose of this note is to give a correct proof of the general case.

On the Thom Isomorphism Theorem

1962 ◽  
Vol 58 (2) ◽  
pp. 206-208 ◽  
Author(s):  
W. H. Cockcroft
Keyword(s):  
Commutative Ring ◽  
Unit Element ◽  
Isomorphism Theorem ◽  
Thom Isomorphism ◽  
Thom Isomorphism Theorem

In what follows (E, p, B) will always denote a fibering in the sense of Serre, † with arc-wise connected base B, projection p:E → B, and fibre F = p−1(b), b ɛ B. Coefficients will always be taken in a commutative ring A with a unit element, and the cohomology will be singular (cubical), as in (3).

scholarly journals On the Unit Graph of a Non-commutative Ring

2015 ◽  
Vol 22 (spec01) ◽  
pp. 817-822 ◽  
Author(s):  
S. Akbari ◽  
E. Estaji ◽  
M.R. Khorsandi
Keyword(s):  
Finite Field ◽  
Commutative Ring ◽  
Bipartite Graph ◽  
Local Ring ◽  
Maximal Ideal ◽  
Complete Bipartite Graph ◽  
Artinian Ring ◽  
Clique Number ◽  
Unit Element ◽  
Unit Graph

Let R be a ring with non-zero identity. The unit graph G(R) of R is a graph with elements of R as its vertices and two distinct vertices a and b are adjacent if and only if a + b is a unit element of R. It was proved that if R is a commutative ring and 𝔪 is a maximal ideal of R such that |R/𝔪| = 2, then G(R) is a complete bipartite graph if and only if (R, 𝔪) is a local ring. In this paper we generalize this result by showing that if R is a ring (not necessarily commutative), then G(R) is a complete r-partite graph if and only if (R, 𝔪) is a local ring and r = |R/𝔪| = 2n for some n ∈ ℕ or R is a finite field. Among other results we show that if R is a left Artinian ring, 2 ∈ U(R) and the clique number of G(R) is finite, then R is a finite ring.

On the Uniqueness of the Coefficient Ring in a Group Ring

1983 ◽  
Vol 35 (4) ◽  
pp. 654-673 ◽  
Author(s):  
Isabelle Adjaero ◽  
Eugene Spiegel
Keyword(s):  
Commutative Ring ◽  
Cyclic Group ◽  
Group Ring ◽  
Commutative Rings ◽  
Infinite Cyclic Group ◽  
Ring Of Polynomials ◽  
Image Position ◽  
Coefficient Ring

Let R1 and R2 be commutative rings with identities, G a group and R1G and R2G the group ring of G over R1 and R2 respectively. The problem that motivates this work is to determine what relations exist between R1 and R2 if R1G and R2G are isomorphic. For example, is the coefficient ring R1 an invariant of R1G? This is not true in general as the following example shows. Let H be a group andIf R1 is a commutative ring with identity and R2 = R1H, thenbut R1 needn't be isomorphic to R2.Several authors have investigated the problem when G = <x>, the infinite cyclic group, partly because of its closeness to R[x], the ring of polynomials over R.

scholarly journals On the natural representation of the symmetric groups

1962 ◽  
Vol 5 (3) ◽  
pp. 121-136 ◽  
Author(s):  
H. K. Farahat
Keyword(s):  
Finite Number ◽  
Symmetric Group ◽  
Commutative Ring ◽  
Group Algebra ◽  
Symmetric Groups ◽  
Unit Element ◽  
Natural Representation ◽  
Image Position ◽  
Obvious Manner

Let E be an arbitrary (non-empty) set and S the restricted symmetric group on E, that is the group of all permutations of E which keep all but a finite number of elements of E fixed. If Φ is any commutative ring with unit element, let Γ = Φ(S) be the group algebra of S over Φ,Γ ⊃ Φ and let M be the free Φ-module having E as Φ-base. The “natural” representation of S is obtained by turning M into a Γ-module in the obvious manner, namely by writing for α∈S, λ1∈Φ,

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