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1.
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
2.
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
3.
A. R. Nasr-Isfahani 《代数通讯》2013,41(3):1302-1320
Let R be a ring with an endomorphism α and an α-derivation δ. In this article, for a skew-Armendariz ring R we study some properties of skew polynomial ring R[x; α, δ]. In particular, among other results, we show that for an (α, δ)-compatible skew-Armendariz ring R, γ(R[x; α, δ]) = γ(R)[x; α, δ] = Ni?*(R)[x; α, δ], where γ is a radical in the class of radicals which includes the Wedderburn, lower nil, Levitzky, and upper nil radicals. We also show that several properties, including the symmetric, reversible, ZCn, zip, and 2-primal property, transfer between R and the skew polynomial ring R[x; α, δ], in case R is (α, δ)-compatible skew-Armendariz. As a consequence we extend and unify several known results. 相似文献
4.
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
5.
6.
A. R. Nasr-Isfahani 《代数通讯》2013,41(2):508-522
Let α be an endomorphism and δ an α-derivation of a ring R. We introduce the notion of skew-Armendariz rings which are a generalization of α-skew Armendariz rings and α-rigid rings and extend the classes of non reduced skew-Armendariz rings. Some properties of this generalization are established, and connections of properties of a skew-Armendariz ring R with those of the Ore extension R[x; α, δ] are investigated. As a consequence we extend and unify several known results related to Armendariz rings. 相似文献
7.
Let α be an endomorphism of R which is not assumed to be surjective and R be α-compatible. It is shown that the skew power series ring R[[x; α]] is right p.q.-Baer if and only if the skew Laurent series ring R[[x, x ?1; α]] is right p.q.-Baer if and only if R is right p.q.-Baer and every countable subset of right semicentral idempotents has a generalized countable join. Examples to illustrate and delimit the theory are provided. 相似文献
8.
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
9.
We first study the quasi-Baerness of R[x; σ, δ] over a quasi-Baer ring R when σ is an automorphism of R, obtaining an affirmative result. We next show that if R is a right principally quasi-Baer ring and σ is an automorphism of R with σ(e) = e for any left semicentral idempotent e ∈ R, then R[x; σ, δ] is right principally quasi-Baer. As a corollary, we have that R[x; δ] over a right principally quasi-Baer ring R is right principally quasi-Baer. Finally, we give conditions under which the quasi-Baernesses (right principal quasi-Baernesses) of R and R[x; σ, δ] are equivalent. 相似文献
10.
Sei-Qwon Oh 《代数通讯》2013,41(10):3007-3012
Let A be a finitely generated Poisson algebra over a field of characteristic zero. Here we prove that every Poisson prime ideal of A is prime and give a method to find all Poisson prime ideals in an arbitrary Poisson polynomial ring A[x; α, δ]. 相似文献
11.
12.
For an endomorphism α of a ring R, we introduce the notion of an α-Armendariz ring to investigate the relative Armendariz properties. This concept extends the class of Armendariz rings and gives us an opportunity to study Armendariz rings in a general setting. It is obvious that every Armendariz ring is an α-Armendariz ring, but we shall give an example to show that there exists a right α-Armendariz ring which is not Armendariz. A number of properties of this version are established. It is shown that if I is a reduced ideal of a ring R such that R/I is a right α-Armendariz ring, then R is right α-Armendariz. For an endomorphism α of a ring R, we show that R is right α-Armendariz if and only if R[x] is right α-Armendariz. Moreover, a weak form of α-Armendariz rings is considered in the last section. We show that in general weak α-Armendariz rings need not be α-Armendariz. 相似文献
13.
Mohammad Habibi 《代数通讯》2017,45(1):151-161
Let R be a ring equipped with an automorphism α and an α-derivation δ. We studied on the relationship between the quasi Baerness and (α, δ)-quasi Baerness of a ring R and these of the inverse skew Laurent series ring R((x?1; α, δ)), in case R is an (α, δ)-weakly rigid ring. Also we proved that for a semicommutative (α, δ)-weakly rigid ring R, R is Baer if and only if so is R((x?1; α, δ)). Moreover for an (α, δ)-weakly rigid ring R, it is shown that the inverse skew Laurent series ring R((x?1; α, δ)) is left p.q.-Baer if and only if R is left p.q.-Baer and every countable subset of left semicentral idempotents of R has a generalized countable join in R. 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.
《代数通讯》2013,41(5):2113-2123
When R is a local ring with a nilpotent maximal ideal, the Ore extension R[x; σ, δ] will or will not be 2-primal depending on the δ-stability of the maximal ideal of R. In the case where R[x; σ, δ] is 2-primal, it will satisfy an even stronger condition; in the case where R[x; σ, δ] is not 2-primal, it will fail to satisfy an even weaker condition. 相似文献
16.
Let R[x; δ] be a differential polynomial ring over a prime Goldie ring R in an indeterminate x, where δ is a derivation of R. In this paper, we describe explicitly the group of δ-stable v-R-ideals and using this results, we show that R[x; δ] is a generalized Asano prime ring if and only if R is a δ-generalized Asano prime ring. 相似文献
17.
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
18.
Evrim Akalan 《代数通讯》2013,41(9):3174-3180
We call a prime Noetherian maximal order R a pseudo-principal ring if every reflexive ideal of R is principal. This class of rings is a broad class properly containing both prime Noetherian pri-(pli) rings and Noetherian unique factorization rings (UFRs). We show that the class of pseudo-principal rings is closed under formation of n × n full matrix rings. Moreover, we prove that if R is a pseudo-principal ring, then the polynomial ring R[x] is also a pseudo-principal ring. We provide examples to illustrate our results. 相似文献
19.
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
20.
Hidetoshi Marubayashi 《Algebras and Representation Theory》2010,13(5):607-622
It is shown that any Ore extension R = V[x;σ,δ] over a total valuation ring V is always v-Bezout which is a generalization of commutative GCD domains. By using this result, a necessary and sufficient
condition are given for R to be fully left bounded. 相似文献