scholarly journals A Generalization of Commutativity Theorem for Rings

2015 ◽  
Vol 61 (1) ◽  
pp. 97-100
Author(s):  
Junchao Wei ◽  
Zhiyong Fan
Keyword(s):  
Commutative Ring ◽  
Commutativity Theorem ◽  
Positive Integers

Abstract Let R be a ring with an identity and for each x ∈ SN(R) = {x ∈ R|x ∉ N(R)} and y ∈ R, (xy)k = xkyk for three consecutive positive integers k. It is shown in this note that R is a commutative ring, which generalizes the known theorem belonging to Ligh and Richoux.

scholarly journals A commutativity theorem for rings

1977 ◽  
Vol 16 (1) ◽  
pp. 75-77 ◽  
Author(s):  
Steve Ligh ◽  
Anthony Richoux
Keyword(s):  
Commutative Ring ◽  
Commutativity Theorem ◽  
Positive Integers

Let R be a ring with an identity and for each x, y in R, (xy)k = xkyk for three consecutive positive integers k. It is shown in this note that R is a commutative ring.

Polynomial and power series ring extensions from sequences

2020 ◽  
pp. 2250048
Author(s):  
Gyu Whan Chang ◽  
Phan Thanh Toan
Keyword(s):  
Power Series ◽  
Commutative Ring ◽  
Trivial Case ◽  
Addition Formula ◽  
Power Series Ring ◽  
Series Ring ◽  
Ring Extensions ◽  
General Sequences ◽  
Divisibility Properties ◽  
Positive Integers

Let [Formula: see text] be a commutative ring with identity. Let [Formula: see text] and [Formula: see text] be the collection of polynomials and, respectively, of power series with coefficients in [Formula: see text]. There are a lot of multiplications in [Formula: see text] and [Formula: see text] such that together with the usual addition, [Formula: see text] and [Formula: see text] become rings that contain [Formula: see text] as a subring. These multiplications are from a class of sequences [Formula: see text] of positive integers. The trivial case of [Formula: see text], i.e. [Formula: see text] for all [Formula: see text], gives the usual polynomial and power series ring. The case [Formula: see text] for all [Formula: see text] gives the well-known Hurwitz polynomial and Hurwitz power series ring. In this paper, we study divisibility properties of these polynomial and power series ring extensions for general sequences [Formula: see text] including UFDs and GCD-domains. We characterize when these polynomial and power series ring extensions are isomorphic to each other. The relation between them and the usual polynomial and power series ring is also presented.

On the Prime Ideals in a Commutative Ring

2000 ◽  
Vol 43 (3) ◽  
pp. 312-319 ◽  
Author(s):  
David E. Dobbs
Keyword(s):  
Prime Ideal ◽  
Commutative Ring ◽  
Polynomial Ring ◽  
Sufficient Conditions ◽  
Prime Ideals ◽  
Preceding Result ◽  
Finite Commutative Ring ◽  
The Axiom Of Choice ◽  
Necessary And Sufficient ◽  
Positive Integers

AbstractIf n and m are positive integers, necessary and sufficient conditions are given for the existence of a finite commutative ring R with exactly n elements and exactly m prime ideals. Next, assuming the Axiom of Choice, it is proved that if R is a commutative ring and T is a commutative R-algebra which is generated by a set I, then each chain of prime ideals of T lying over the same prime ideal of R has at most 2|I| elements. A polynomial ring example shows that the preceding result is best-possible.

A multiplicative property of R-sequences and H1-sets

1975 ◽  
Vol 78 (1) ◽  
pp. 1-6 ◽  
Author(s):  
C. P. L. Rhodes
Keyword(s):  
Commutative Ring ◽  
Multiplicative Property ◽  
Rees Algebras ◽  
Image Position ◽  
Positive Integers ◽  
The Ideal

Let R be a commutative ring which may not contain a multiplicative identity. A set of elements a1,…,ak in R will be called an H1-set (this notation is explained in section 1) if for each relation r1a1 + … +rkak = 0 (ri ∈ R) there exist elements sij ∈ R such thatwhere Xl,…,Xk are indeterminates. Any R-sequence is an H1-set, but there do exist H1-sets which are not R-sequences (see section 1). Throughout this note we consider an H1-set a1,…,ak which we suppose to be partitioned into two non-empty sets bl…, br and cl,…, cs. Our main purpose is to show that the ideals B = Rb1 + … + Rbr and C = Rc1 + … + Rcs satisfy Bm ∩ Cn = BmCn for all positive integers m and n (Corollary 1). This generalizes Lemma 2 of Caruth(2) where the result is proved when a1,…, ak is a permutable R-sequence. Our proof involves more detail than is necessary just for this, and we obtain various other properties of H1-sets. In particular we extend the main results of Corsini(3) concerning the symmetric and Rees algebras of a power of the ideal Ra1 +… + Rak (Corollary 3).

scholarly journals On asymptotic values of certain sets of attached prime ideals

1988 ◽  
Vol 30 (3) ◽  
pp. 293-300 ◽  
Author(s):  
A.-J. Taherizadeh
Keyword(s):  
Commutative Ring ◽  
Noetherian Ring ◽  
Prime Ideals ◽  
Primary Decomposition ◽  
Prime Divisors ◽  
Finitely Generated Module ◽  
Commutative Noetherian Ring ◽  
Asymptotic Values ◽  
Attached Prime ◽  
Positive Integers

In his paper [1], M. Brodmann showed that if M is a1 finitely generated module over the commutative Noetherian ring R (with identity) and a is an ideal of R then the sequence of sets {Ass(M/anM)}n∈ℕ and {Ass(an−1M/anM)}n∈ℕ (where ℕ denotes the set of positive integers) are eventually constant. Since then, the theory of asymptotic prime divisors has been studied extensively: in [5], Chapters 1 and 2], for example, various results concerning the eventual stable values of Ass(R/an;) and Ass(an−1/an), denoted by A*(a) and B*(a) respectively, are discussed. It is worth mentioning that the above mentioned results of Brodmann still hold if one assumes only that A is a commutative ring (with identity) and M is a Noetherian A-module, and AssA(M), in this situation, is regarded as the set of prime ideals belonging to the zero submodule of M for primary decomposition.

scholarly journals On Laplacian Eigenvalues of the Zero-Divisor Graph Associated to the Ring of Integers Modulo n

Mathematics ◽  
2021 ◽  
Vol 9 (5) ◽  
pp. 482
Author(s):  
Bilal A. Rather ◽  
Shariefuddin Pirzada ◽  
Tariq A. Naikoo ◽  
Yilun Shang
Keyword(s):  
Commutative Ring ◽  
Zero Divisor ◽  
Simple Graph ◽  
Laplacian Spectrum ◽  
Laplacian Eigenvalues ◽  
Ring Of Integers ◽  
Zero Divisor Graph ◽  
Zero Divisors ◽  
Vertex Set ◽  
Positive Integers

Given a commutative ring R with identity 1≠0, let the set Z(R) denote the set of zero-divisors and let Z*(R)=Z(R)∖{0} be the set of non-zero zero-divisors of R. The zero-divisor graph of R, denoted by Γ(R), is a simple graph whose vertex set is Z*(R) and each pair of vertices in Z*(R) are adjacent when their product is 0. In this article, we find the structure and Laplacian spectrum of the zero-divisor graphs Γ(Zn) for n=pN1qN2, where p<q are primes and N1,N2 are positive integers.

scholarly journals On Fully (m,n)-stable modules relative to an ideal A of

2015 ◽  
Vol 12 (2) ◽  
pp. 400-405
Author(s):  
Baghdad Science Journal
Keyword(s):  
Commutative Ring ◽  
Identity Element ◽  
Characterization Theorems ◽  
Positive Integers

Let R be a commutative ring with non-zero identity element. For two fixed positive integers m and n. A right R-module M is called fully (m,n) -stable relative to ideal A of , if for each n-generated submodule of Mm and R-homomorphism . In this paper we give some characterization theorems and properties of fully (m,n) -stable modules relative to an ideal A of . which generalize the results of fully stable modules relative to an ideal A of R.

scholarly journals A commutativity theorem for semi-primitive rings

1986 ◽  
Vol 34 (2) ◽  
pp. 293-295 ◽  
Author(s):  
Hisao Tominaga
Keyword(s):  
Commutativity Theorem ◽  
Positive Integers ◽  
Primitive Ring

In this brief note, we prove the following: Let R be a semi-primitive ring. Suppose that for each pair x, y ε R there exist positive integers m = m (x,y) and n = n (x,y) such that either [xm,(xy) n − (yx) n] = 0 or [xm,(xy) n + (yx) n] = 0. Then R is commutative.

scholarly journals A note on reductions of ideals relative to an Artinian module

1993 ◽  
Vol 35 (2) ◽  
pp. 219-224 ◽  
Author(s):  
A.-J. Taherizadeh
Keyword(s):  
Positive Integer ◽  
Commutative Ring ◽  
Integral Closure ◽  
Artinian Modules ◽  
Artinian Module ◽  
Image Position ◽  
Positive Integers

The concept of reduction and integral closure of ideals relative to Artinian modules were introduced in [7]; and we summarize some of the main aspects now.Let A be a commutative ring (with non-zero identity) and let a, b be ideals of A. Suppose that M is an Artinian module over A. We say that a is a reduction of b relative to M if a ⊆ b and there is a positive integer s such that)O:Mabs)=(O:Mbs+l).An element x of A is said to be integrally dependent on a relative to M if there exists n y ℕ(where ℕ denotes the set of positive integers) such thatIt is shown that this is the case if and only if a is a reduction of a+Ax relative to M; moreoverᾱ={x ɛ A: xis integrally dependent on a relative to M}is an ideal of A called the integral closure of a relative to M and is the unique maximal member of℘ = {b: b is an ideal of A which has a as a reduction relative to M}.

Total graph of the ring ℤn × ℤm

2015 ◽  
Vol 07 (01) ◽  
pp. 1550004 ◽  
Author(s):  
Alpesh M. Dhorajia
Keyword(s):  
Commutative Ring ◽  
Undirected Graph ◽  
Clique Number ◽  
Total Graph ◽  
Fundamental Properties ◽  
Zero Divisors ◽  
Independent Number ◽  
Positive Integers

Let R be a commutative ring and Z(R) be the set of all zero-divisors of R. The total graph of R, denoted by T Γ(R), is the (undirected) graph with vertices set R. For any two distinct elements x, y ∈ R, the vertices x and y are adjacent if and only if x + y ∈ Z(R). In this paper, we obtain certain fundamental properties of the total graph of ℤn × ℤm, where n and m are positive integers. We determine the clique number and independent number of the total graph T Γ(ℤn × ℤm).

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