共查询到20条相似文献,搜索用时 15 毫秒
1.
Nathan Bloomfield 《代数通讯》2013,41(2):765-775
To each commutative ring R we can associate a zero divisor graph whose vertices are the zero divisors of R and such that two vertices are adjacent if their product is zero. Detecting isomorphisms among zero divisor graphs can be reduced to the problem of computing the classes of R under a suitable semigroup congruence. Presently, we introduce a strategy for computing this quotient for local rings using knowledge about a generating set for the maximal ideal. As an example, we then compute Γ(R) for several classes of rings; with the results in [4] these classes include all local rings of order p 4 and p 5 for prime p. 相似文献
2.
We study the rigid property of rings in the concrete, via power series rings, matrix rings, and Insertion-of-Factors-Property. In the procedure, we concentrate our attention on the skew power-serieswise Armendariz property. We next apply the McCoy condition to skew power series rings, which induces a generalization of skew power-serieswise Armendariz. The relationship is investigated among the properties above and related ring properties, and several known results are obtained as consequences of our results. 相似文献
3.
Mi Hee Park 《代数通讯》2015,43(1):51-58
Let R be an m-dimensional pseudo-valuation domain with residue field k, let V be the associated valuation domain with residue field K, and let k 0 be the maximal separable extension of k in K. We compute the t-dimension of polynomial and power series rings over R. It is easy to see that t-dim R[x 1,…, x n ] = 2 if m = 1 and K is transcendental over k, but equals m otherwise, and that t-dim R[[x 1,…, x n ]] = ∞ if R is a nonSFT-ring. When R is an SFT-ring, we also show that: (1) t-dim R[[x]] = m; (2) t-dim R[[x 1,…, x n ]] = 2m ? 1, if n ≥ 2, K has finite exponent over k 0, and [k 0: k] < ∞; (3) t-dim R[[x 1,…, x n ]] = 2m, otherwise. 相似文献
4.
Sandra Spiroff 《代数通讯》2013,41(7):2338-2348
We study the zero divisor graph determined by equivalence classes of zero divisors of a commutative Noetherian ring R. We demonstrate how to recover information about R from this structure. In particular, we determine how to identify associated primes from the graph. 相似文献
5.
For a ring endomorphism α, we introduce and investigate SPA-rings which are a generalization of α-rigid rings and determine the radicals of the skew polynomial rings R[x; α], R[x, x ?1; α] and the skew power series rings R[[x; α]], R[[x, x ?1; α]], in terms of those of R. We prove that several properties transfer between R and the extensions, in case R is an SPA-ring. We will construct various types of nonreduced SPA-rings and show SPA is a strictly stronger condition than α-rigid. 相似文献
6.
Let R be a ring, S a strictly ordered monoid, and ω: S → End(R) a monoid homomorphism. In [30], Marks, Mazurek, and Ziembowski study the (S, ω)-Armendariz condition on R, a generalization of the standard Armendariz condition from polynomials to skew generalized power series. Following [30], we provide various classes of nonreduced (S, ω)-Armendariz rings, and determine radicals of the skew generalized power series ring R[[S ≤, ω]], in terms of those of an (S, ω)-Armendariz ring R. We also obtain some characterizations for a skew generalized power series ring to be local, semilocal, clean, exchange, uniquely clean, 2-primal, or symmetric. 相似文献
7.
Based on a theorem of McCoy on commutative rings, Nielsen called a ring R right McCoy if, for any nonzero polynomials f(x), g(x) over R, f(x)g(x) = 0 implies f(x)r = 0 for some 0 ≠ r ? R. In this note, we consider a skew version of these rings, called σ-skew McCoy rings, with respect to a ring endomorphism σ. When σ is the identity endomorphism, this coincides with the notion of a right McCoy ring. Basic properties of σ-skew McCoy rings are observed, and some of the known results on right McCoy rings are obtained as corollaries. 相似文献
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9.
We introduce and study subrings with simple 0-multiplication of matrix rings in the context of Armendariz rings. In this way we extend several known results in the area. 相似文献
10.
Mason introduced the reflexive property for ideals, and then this concept was generalized by Kim and Baik, defining idempotent reflexive right ideals and rings. In this article, we characterize aspects of the reflexive and one-sided idempotent reflexive properties, showing that the concept of idempotent reflexive ring is not left-right symmetric. It is proved that a (right idempotent) reflexive ring which is not semiprime (resp., reflexive), can always be constructed from any semiprime (resp., reflexive) ring. It is also proved that the reflexive condition is Morita invariant and that the right quotient ring of a reflexive ring is reflexive. It is shown that both the polynomial ring and the power series ring over a reflexive ring are idempotent reflexive. We obtain additionally that the semiprimeness, reflexive property and one-sided idempotent reflexive property of a ring coincide for right principally quasi-Baer rings. 相似文献
11.
广义幂级数环上的PS模 总被引:1,自引:0,他引:1
Let R be a commutative ring and(S,≤)a strictly totally ordered monoid which satisfies the condition that 0≤s for every s ∈ S,In this paper we show that if RM is a PS-module,then the module [[M^s,≤]]of generalized power series over M is a PS [[R^s,≤]]-module. 相似文献
12.
平坦的多项式剩余类环 总被引:1,自引:0,他引:1
本文证明了如果多项式的剩余类环 A=R[T]/fR[T]作为 R-模是平坦模,且R是约化环,则f是正规多项式.特别地,若R还是连通的,则f的首项系数是单位.也证明了弱整体有限的凝聚环是约化环,以及弱整体为有限的凝聚连通环是整环. 相似文献
13.
Let R be a ring, (S, ≤) a strictly ordered monoid and ω: S → End(R) a monoid homomorphism. The skew generalized power series ring R[[S, ω]] is a common generalization of (skew) polynomial rings, (skew) power series rings, (skew) Laurent polynomial rings, (skew) group rings, and Mal'cev–Neumann Laurent series rings. In this article, we study relations between the (quasi-) Baer, principally quasi-Baer and principally projective properties of a ring R, and its skew generalized power series extension R[[S, ω]]. As particular cases of our general results, we obtain new theorems on (skew) group rings, Mal'cev–Neumann Laurent series rings, and the ring of generalized power series. 相似文献
14.
Cameron Wickham 《代数通讯》2013,41(2):325-345
This paper investigates properties of the zero-divisor graph of a commutative ring and its genus. In particular, we determine all isomorphism classes of finite commutative rings with identity whose zero-divisor graph has genus one. 相似文献
15.
斜幂级数环的主拟Baer性 总被引:4,自引:0,他引:4
设R是环,并且R的左半中心幂等元都是中心幂等元, α是R的一个弱刚性自同态. 本文证明了斜幂级数环R[[x,α]]是右主拟Baer环当且仅当R是右主拟Baer环,并且R的任意可数幂等元集在I(R)中有广义交,其中I(R)是R的幂等元集. 相似文献
16.
Gyu Whan Chang 《代数通讯》2013,41(10):4182-4187
Let α be an infinite cardinal number, Λ be an index set of cardinality > α, and {X λ}λ∈Λ be a set of indeterminates over an integral domain D. It is well known that there are three ways of defining the ring of formal power series in {X λ}λ∈Λ over D, say, D[[{X λ}]] i for i = 1, 2, 3. In this paper, we let D[[{X λ}]]α = ∪ {D[[{X λ}λ∈Γ]]3 | Γ ? Λ and |Γ| ≤ α}, and we then show that D[[{X λ}]]α is an integral domain such that D[[{X λ}]]2 ? D[[{X λ}]]α ? D[[{X λ}]]3. We also prove that (1) D is a Krull domain if and only if D[[{X λ}]]α is a Krull domain and (2) D[[{X λ}]]α is a unique factorization domain (UFD) (resp., π-domain) if and only if D[[X 1,…, X n ]] is a UFD (resp., π-domain) for every integer n ≥ 1. 相似文献
17.
A ring R is called an M-quasi-Armendariz ring (a quasi-Armendariz ring relative to a monoid M) if whenever elements α = a 1 g 1 + a 2 g 2 + ··· + a n g n , β = b 1 h 1 + b 2 h 2 + ··· + b m h m ? R[M] satisfy α R[M]β = 0, then a i Rb j = 0 for each i, j. After discussing some basic properties of M-quasi-Armendariz rings, we consider the influence of transformation of the monoid M and the ring R on this property. Particularly, we give some sufficient conditions for the monoids M, N, and the ring R under which R is M × N-quasi-Armendariz if and only if R is M-quasi-Armendariz and N-quasi-Armendariz. 相似文献
18.
Abdullah M. Abdul-Jabbar Chenar Abdul Kareem Ahmed Yang Lee Young Joo Seo 《代数通讯》2017,45(11):4881-4895
The reversible property is an important role in noncommutative ring theory. Recently, the study of the reversible ring property on nilpotent elements is established by Abdul-Jabbar et al., introducing the concept of commutativity of nilpotent elements at zero (simply, a CNZ ring) as a generalization of reversible rings. We here study this property skewed by a ring endomorphism α, and such ring is called a right α-skew CNZ ring which is an extension of CNZ rings as well as a generalization of right α-skew reversible rings, and then investigate the structure of right α-skew CNZ rings and their related properties. Consequently, several known results are obtained as corollaries of our results. 相似文献
19.
Chen Huanyin 《东北数学》1997,(2)
PolynomialRingsoverNon┐noetherianPowerSeriesRingsChenHuanyin(陈焕艮)(DepartmentofMathematics,HunanNormalUniversity,Changsha,4100... 相似文献
20.