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1.
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
2.
Blake W. Madill 《代数通讯》2013,41(3):913-918
Let R be a ring satisfying a polynomial identity, and let D be a derivation of R. We consider the Jacobson radical of the skew polynomial ring R[x; D] with coefficients in R and with respect to D, and show that J(R[x; D]) ∩ R is a nil D-ideal. This extends a result of Ferrero, Kishimoto, and Motose, who proved this in the case when R is commutative. 相似文献
3.
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
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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 paper we obtain necessary and sufficient conditions for the skew generalized power series ring R[[S,ω]] to be a semiprime, prime, quasi-Baer, or Baer ring. Furthermore, we study the prime radical of a skew generalized power series ring R[[S,ω]]. Our results extend and unify many existing results. In particular, 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
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A. R. Nasr-Isfahani 《代数通讯》2013,41(3):1337-1349
In this note we study radicals of skew polynomial ring R[x; α] and skew Laurent polynomial ring R[x, x ?1; α], for a skew-Armendariz ring R. In particular, among the other results, we show that for an skew-Armendariz ring R, J(R[x; α]) = N 0(R[x; α]) = Ni?*(R)[x; α] and J(R[x, x ?1; α]) = N 0(R[x, x ?1; α]) = Ni?*(R)[x, x ?1; α]. 相似文献
7.
We study the skew inverse Laurent-serieswise Armendariz (or simply, SIL-Armendariz) condition on R, a generalization of the standard Armendariz condition from polynomials to skew inverse Laurent series. We study relations between the set of annihilators in R and the set of annihilators in R((x ?1; α)). Among applications, we show that a number of interesting properties of a SIL-Armendariz ring R such as the Baer and the α-quasi Baer property transfer to its skew inverse Laurent series extensions R((x ?1; α)) and vice versa. For an α-weakly rigid ring R, R((x ?1; α)) is a left p.q.-Baer ring if and only if R is left p.q.-Baer and every countable subset of S ?(R) has a generalized countable join in R. Various types of examples of SIL-Armendariz rings is provided. 相似文献
8.
Michał Ziembowski 《代数通讯》2013,41(2):664-666
One of the main results of the article [2] says that, if a ring R is semiperfect and ? is an authomorphism of R, then the skew Laurent series ring R((x, ?)) is semiperfect. We will show that the above statement is not true. More precisely, we will show that, if the Laurent series ring R((x)) is semilocal, then R is semiperfect with nil Jacobson radical. 相似文献
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AbstractIt is well known that when a ring R satisfies ACC on right annihilators of elements, then the right singular ideal of R is nil, in this case, we say R is right nil-singular. Many classes of rings whose singular ideals are nil, but do not satisfy the ACC on right annihilators, are presented and the behavior of them is investigated with respect to various constructions, in particular skew polynomial rings and triangular matrix rings. The class of right nil-singular rings contains π-regular rings and is closed under direct sums. Examples are provided to explain and delimit our results. 相似文献
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Yi Zhong 《数学学报(英文版)》1997,13(4):433-442
In this paper we prove that under some natural conditions, the Ore extensions and skew Laurent polynomial rings are injectively
homogeneous or homologically homogeneous if so are their coefficient rings. Specifically, we prove that ifR is a commutative Noetherian ring of positive characteristic, thenA
n (R), then
th Weyl algebra overR, is injectively homogeneous (resp. homologically homogeneous) ifR has finite injective dimension (resp. global dimension). 相似文献
12.
We prove that if R is a semiprime ring and α is a partial action of an infinite cyclic group on R, then R is right Goldie if and only if R[x; α] is right Goldie if and only if R?x; α? is right Goldie, where R[x; α] (R?x; α?) denotes the partial skew (Laurent) polynomial ring over R. In addition, R?x; α? is semiprime while R[x; α] is not necessarily semiprime. 相似文献
13.
Huanyin Chen 《代数通讯》2013,41(10):3790-3804
An element of a ring is called strongly J-clean provided that it can be written as the sum of an idempotent and an element in its Jacobson radical that commute. A ring is strongly J-clean in case each of its elements is strongly J-clean. We investigate, in this article, strongly J-clean rings and ultimately deduce strong J-cleanness of T n (R) for a large class of local rings R. Further, we prove that the ring of all 2 × 2 matrices over commutative local rings is not strongly J-clean. For local rings, we get criteria on strong J-cleanness of 2 × 2 matrices in terms of similarity of matrices. The strong J-cleanness of a 2 × 2 matrix over commutative local rings is completely characterized by means of a quadratic equation. 相似文献
14.
Let R be a ring with identity. The polynomial ring over R is denoted by R[x] with x its indeterminate. It is shown that polynomial rings over symmetric rings need not be symmetric by an example. 相似文献
15.
ABSTRACT In this paper, we study the behavior of the couniform (or dual Goldie) dimension of a module under various polynomial extensions. For a ring automorphism σ ∈ Aut(R), we use the notion of a σ-compatible module M R to obtain results on the couniform dimension of the polynomial modules M[x], M[x ?1], and M[x, x ?1] over suitable skew extension rings. 相似文献
16.
Abdollah Alhevaz 《代数通讯》2017,45(3):919-923
Ever since the introduction, skew inverse Laurent series rings have kept growing in importance, as researchers characterized their properties (such as Noetherianness, Armendarizness, McCoyness, etc.) in terms of intrinsic properties of the base ring and studied their relations with other fields of mathematics, as for example quantum mechanics theory. The goal of our paper is to study the primeness and semiprimeness of general skew inverse Laurent series rings R((x?1;σ,δ)), where R is an associative ring equipped with an automorphism σ and a σ-derivation δ. 相似文献
17.
An R-module M is called strongly duo if Tr(N, M) = N for every N ≤ M R . Several equivalent conditions to being strongly duo are given. If M R is strongly duo and reduced, then End R (M) is a strongly regular ring and the converse is true when R is a Dedekind domain and M R is torsion. Over certain rings, nonsingular strongly duo modules are precisely regular duo modules. If R is a Dedekind domain, then M R is strongly duo if and only if either M ≈ R or M R is torsion and duo. Over a commutative ring, strongly duo modules are precisely pq-injective duo modules and every projective strongly duo module is a multiplication module. A ring R is called right strongly duo if R R is strongly duo. Strongly regular rings are precisely reduced (right) strongly duo rings. A ring R is Noetherian and all of its factor rings are right strongly duo if and only if R is a serial Artinian right duo ring. 相似文献
18.
A. R. Nasr-Isfahani 《代数通讯》2013,41(11):4461-4469
For a ring R, endomorphism α of R and positive integer n we define a skew triangular matrix ring T n (R, α). By using an ideal theory of a skew triangular matrix ring T n (R, α) we can determine prime, primitive, maximal ideals and radicals of the ring R[x; α]/ ? x n ?, for each positive integer n, where R[x; α] is the skew polynomial ring, and ? x n ? is the ideal generated by x n . 相似文献
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