SKEW POWER SERIES RINGS OF DERIVATION TYPE

2011 ◽  
Vol 10 (06) ◽  
pp. 1383-1399 ◽  
Author(s):  
JEFFREY BERGEN ◽  
PIOTR GRZESZCZUK

In this paper, we contrast the structure of a noncommutative algebra R with that of the skew power series ring R[[y;d]]. Several of our main results examine when the rings R, Rd, and R[[y;d]] are prime or semiprime under the assumption that d is a locally nilpotent derivation.

1995 ◽  
Vol 38 (4) ◽  
pp. 429-433 ◽  
Author(s):  
David E. Dobbs ◽  
Moshe Roitman

AbstractIt is proved that if r* is the weak normalization of an integral domain r, then the weak normalization of the power series ring r[[x1,....xn]] is contained in R*[[X1,....Xn]]. Consequently, if R is a weakly normal integral domain, then R[[X1,....Xn]] is also weakly normal.


1986 ◽  
Vol 38 (1) ◽  
pp. 158-178 ◽  
Author(s):  
Paul Roberts

A common method in studying a commutative Noetherian local ring A is to find a regular subring R contained in A so that A becomes a finitely generated R-module, and in this way one can obtain some information about the original ring by applying what is known about regular local rings. By the structure theorems of Cohen, if A is complete and contains a field, there will always exist such a subring R, and R will be a power series ring k[[X1, …, Xn]] = k[[X]] over a field k. In this paper we show that if R is chosen properly, the ring A (or, more generally, an A-module M), will have a comparatively simple structure as an R-module. More precisely, A (or M) will have a free resolution which resembles the Koszul complex on the variables (X1, …, Xn) = (X); such a complex will be called an (X)-graded complex and will be given a precise definition below.


2015 ◽  
Vol 25 (05) ◽  
pp. 725-744 ◽  
Author(s):  
Ryszard Mazurek ◽  
Michał Ziembowski

Let R be a ring, and let S be a strictly ordered monoid. The generalized power series ring R[[S]] is a common generalization of polynomial rings, Laurent polynomial rings, power series rings, Laurent series rings, Mal'cev–Neumann series rings, monoid rings and group rings. In this paper, we examine which conditions on R and S are necessary and which are sufficient for the generalized power series ring R[[S]] to be semilocal right Bézout or semilocal right distributive. In the case where S is a strictly totally ordered monoid we characterize generalized power series rings R[[S]] that are semilocal right distributive or semilocal right Bézout (the latter under the additional assumption that S is not a group).


2013 ◽  
Vol 13 (02) ◽  
pp. 1350083 ◽  
Author(s):  
A. ALHEVAZ ◽  
D. KIANI

One of the important properties of commutative rings, proved by McCoy [Remarks on divisors of zero, Amer. Math. Monthly49(5) (1942) 286–295], is that if two nonzero polynomials annihilate each other over a commutative ring then each polynomial has a nonzero annihilator in the base ring. Nielsen [Semi-commutativity and the McCoy condition, J. Algebra298(1) (2006) 134–141] generalizes this property to non-commutative rings. Let M be a monoid and σ be an automorphism of a ring R. For the continuation of McCoy property of non-commutative rings, in this paper, we extend the McCoy's theorem to skew Laurent power series ring R[[x, x-1; σ]] and skew monoid ring R * M over general non-commutative rings. Constructing various examples, we classify how these skew versions of McCoy property behaves under various ring extensions. Moreover, we investigate relations between these properties and other standard ring-theoretic properties such as zip rings and rings with Property (A). As a consequence we extend and unify several known results related to McCoy rings.


Author(s):  
Refaat M. Salem ◽  
Mohamed A. Farahat ◽  
Hanan Abd-Elmalk

A rightR-moduleMRis called a PS-module if its socle,SocMR, is projective. We investigate PS-modules over Ore extension and skew generalized power series extension. LetRbe an associative ring with identity,MRa unitary rightR-module,O=Rx;α,δOre extension,MxOa rightO-module,S,≤a strictly ordered additive monoid,ω:S→EndRa monoid homomorphism,A=RS,≤,ωthe skew generalized power series ring, andBA=MS,≤RS,≤, ωthe skew generalized power series module. Then, under some certain conditions, we prove the following: (1) IfMRis a right PS-module, thenMxOis a right PS-module. (2) IfMRis a right PS-module, thenBAis a right PS-module.


2013 ◽  
Vol 50 (4) ◽  
pp. 436-453
Author(s):  
A. Majidinya ◽  
A. Moussavi

By [12], a ring R is left APP if R has the property that “the left annihilator of a principal ideal is pure as a left ideal”. Equivalently, R is a left APP-ring if R modulo the left annihilator of any principal left ideal is flat. Let R be a ring, (S, ≦) a strictly totally ordered commutative monoid and ω: S → End(R) a monoid homomorphism. Following [16], we show that, when R is a (S, ω)-weakly rigid and (S, ω)-Armendariz ring, then the skew generalized power series ring R[[S≦, ω]] is right APP if and only if rR(A) is S-indexed left s-unital for every S-indexed generated right ideal A of R. We also show that when R is a (S, ω)-strongly Armendariz ring and ω(S) ⫅ Aut(R), then the ring R[[S≦, ω]] is left APP if and only if ℓR(∑a∈A ∑s∈SRωs(a)) is S-indexed right s-unital, for any S-indexed subset A of R. In particular, when R is Armendariz relative to S, then R[[S≦]] is right APP if and only if rR(A) is S-indexed left s-unital, for any S-indexed generated right ideal A of R.


2016 ◽  
Vol 10 (02) ◽  
pp. 1750034 ◽  
Author(s):  
Kamal Paykan ◽  
Ahmad Moussavi

Let [Formula: see text] be a ring, [Formula: see text] a strictly ordered monoid, and [Formula: see text] a monoid homomorphism. The skew generalized power series ring [Formula: see text] 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 introduce and study the [Formula: see text]-nil-Armendariz condition on [Formula: see text], a generalization of the standard nil-Armendariz condition from polynomials to skew generalized power series. We resolve the structure of [Formula: see text]-nil-Armendariz rings and obtain various necessary or sufficient conditions for a ring to be [Formula: see text]-nil-Armendariz. The [Formula: see text]-nil-Armendariz condition is connected to the question of whether or not a skew generalized power series ring [Formula: see text] over a nil ring [Formula: see text] is nil, which is related to a question of Amitsur [Algebras over infinite fields, Proc. Amer. Math. Soc. 7 (1956) 35–48]. As particular cases of our general results we obtain several new theorems on the nil-Armendariz condition. Our results extend and unify many existing results.


2014 ◽  
Vol 13 (07) ◽  
pp. 1450048 ◽  
Author(s):  
Ryszard Mazurek

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) Laurent polynomial rings, (skew) power series rings, (skew) Laurent series rings, (skew) monoid rings, (skew) Mal'cev–Neumann series rings, and generalized power series rings. We characterize those subsets T of S for which the cut-off operator with respect to T is a Rota–Baxter operator on the ring R[[S, ω]]. The obtained results provide a large class of noncommutative Rota–Baxter algebras.


Author(s):  
A. Moussavi

Let [Formula: see text] be a monomorphism of a ring [Formula: see text] which is not assumed to be surjective. It is shown that, for an [Formula: see text]-weakly rigid [Formula: see text], the skew power series ring [Formula: see text] is right p.q.-Baer if and only if the skew Laurent series ring [Formula: see text] is right p.q.-Baer if and only if [Formula: see text] is right p.q.-Baer and every countable subset of right semicentral idempotents has a generalized countable join.


2012 ◽  
Vol 12 (01) ◽  
pp. 1250129 ◽  
Author(s):  
A. R. NASR-ISFAHANI

Let R be a ring, (S, ≤) a strictly ordered monoid and ω : S → End (R) a monoid homomorphism. In this note for a (S, ω)-Armendariz ring R we study some properties of skew generalized power series ring R[[S, ω]]. In particular, among other results, we show that for a S-compatible (S, ω)-Armendariz ring R, α(R[[S, ω]]) = α(R)[[S, ω]] = Ni ℓ*(R)[[S, ω]], where α is a radical in a 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 generalized power series ring R[[S, ω]], in case R is S-compatible (S, ω)-Armendariz.


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