THE IMPACT OF THE MONOID HOMOMORPHISM ON THE STRUCTURE OF SKEW GENERALIZED POWER SERIES RINGS

2018 ◽  
Vol 103 (7) ◽  
pp. 1215-1227
Author(s):  
Ahmad Faisol ◽  
Budi Surodjo ◽  
Sri Wahyuni
Keyword(s):  
Power Series ◽  
Generalized Power Series ◽  
Monoid Homomorphism ◽  
Power Series Rings ◽  
The Impact

scholarly journals PS-Modules over Ore Extensions and Skew Generalized Power Series Rings

2015 ◽  
Vol 2015 ◽  
pp. 1-6
Author(s):  
Refaat M. Salem ◽  
Mohamed A. Farahat ◽  
Hanan Abd-Elmalk
Keyword(s):  
Power Series ◽  
Associative Ring ◽  
Ore Extension ◽  
Power Series Ring ◽  
Generalized Power Series ◽  
Series Ring ◽  
Monoid Homomorphism ◽  
Ore Extensions ◽  
Power Series Rings ◽  
Unitary Right

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.

On app skew generalized power series rings

2013 ◽  
Vol 50 (4) ◽  
pp. 436-453
Author(s):  
A. Majidinya ◽  
A. Moussavi
Keyword(s):  
Power Series ◽  
Left Ideal ◽  
Commutative Monoid ◽  
Power Series Ring ◽  
Generalized Power Series ◽  
Series Ring ◽  
Monoid Homomorphism ◽  
Power Series Rings ◽  
Armendariz Ring ◽  
Left Annihilator

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.

Nilpotent elements and nil-Armendariz property of skew generalized power series rings

2016 ◽  
Vol 10 (02) ◽  
pp. 1750034 ◽  
Author(s):  
Kamal Paykan ◽  
Ahmad Moussavi
Keyword(s):  
Power Series ◽  
Sufficient Conditions ◽  
Polynomial Rings ◽  
Group Rings ◽  
Power Series Ring ◽  
Generalized Power Series ◽  
Series Ring ◽  
Skew Polynomial Rings ◽  
Monoid Homomorphism ◽  
Power Series Rings

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.

Rota–Baxter operators on skew generalized power series rings

2014 ◽  
Vol 13 (07) ◽  
pp. 1450048 ◽  
Author(s):  
Ryszard Mazurek
Keyword(s):  
Power Series ◽  
Neumann Series ◽  
Polynomial Rings ◽  
Power Series Ring ◽  
Generalized Power Series ◽  
Series Ring ◽  
Skew Polynomial Rings ◽  
Monoid Homomorphism ◽  
Power Series Rings ◽  
Skew Polynomial

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.

RADICALS OF SKEW GENERALIZED POWER SERIES RINGS

2012 ◽  
Vol 12 (01) ◽  
pp. 1250129 ◽  
Author(s):  
A. R. NASR-ISFAHANI
Keyword(s):  
Power Series ◽  
Power Series Ring ◽  
Generalized Power Series ◽  
Series Ring ◽  
Ordered Monoid ◽  
Monoid Homomorphism ◽  
Property Transfer ◽  
Power Series Rings ◽  
Armendariz Ring

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.

Unified Extensions of Strongly Reversible Rings and Links with Other Classic Ring Theoretic Properties

2018 ◽  
Vol 85 (3-4) ◽  
pp. 434
Author(s):  
R. K. Sharma ◽  
Amit B. Singh
Keyword(s):  
Power Series ◽  
Polynomial Rings ◽  
Group Rings ◽  
Laurent Polynomials ◽  
Power Series Ring ◽  
Generalized Power Series ◽  
Series Ring ◽  
Skew Polynomial Rings ◽  
Monoid Homomorphism ◽  
Power Series Rings

Let R be a ring, (M, ≤) a strictly ordered monoid and ω : M → <em>End</em>(R) a monoid homomorphism. The skew generalized power series ring R[[M; ω]] is a compact generalization of (skew) polynomial rings, (skew) power series rings, (skew) Laurent polynomials rings, (skew) Laurent power series rings, (skew) group rings, (skew) monoid rings, Mal'cev Neumann rings and generalized power series rings. In this paper, we introduce concept of strongly (M, ω)-reversible ring (strongly reversible ring related to skew generalized power series ring R[[M, ω]]) which is a uni ed generalization of strongly reversible ring and study basic properties of strongly (M; ω)-reversible. The Nagata extension of strongly reversible is proved to be strongly reversible if R is Armendariz. Finally, it is proved that strongly reversible ring strictly lies between reduced and reversible ring in the expanded diagram given by Diesl et. al. [7].

Left principally quasi-Baer and left APP-rings of skew generalized power series

2014 ◽  
Vol 14 (03) ◽  
pp. 1550038 ◽  
Author(s):  
Ryszard Mazurek
Keyword(s):  
Power Series ◽  
Polynomial Rings ◽  
Power Series Ring ◽  
Generalized Power Series ◽  
Series Ring ◽  
Skew Polynomial Rings ◽  
Monoid Homomorphism ◽  
Power Series Rings ◽  
Skew Polynomial ◽  
Classical Ring

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, and (skew) monoid rings. We characterize when a skew generalized power series ring R[[S, ω]] is left principally quasi-Baer and under various finiteness conditions on R we characterize when the ring R[[S, ω]] is left APP. As immediate corollaries we obtain characterizations for all aforementioned classical ring constructions to be left principally quasi-Baer or left APP. Such a general approach not only gives new results for several constructions simultaneously, but also serves the unification of already known results.

scholarly journals The Ring Homomorphisms of Matrix Rings over Skew Generalized Power Series Rings

CAUCHY ◽  
2021 ◽  
Vol 7 (1) ◽  
pp. 129-135
Author(s):  
Ahmad Faisol ◽  
Fitriani Fitriani
Keyword(s):  
Power Series ◽  
Identity Element ◽  
Sufficient Conditions ◽  
Commutative Rings ◽  
Ring Homomorphism ◽  
Matrix Rings ◽  
Generalized Power Series ◽  
Ring Homomorphisms ◽  
Monoid Homomorphism ◽  
Power Series Rings

Let  M_n (R_1 [[S_1,≤_1,ω_1]]) and M_n (R_2 [[S_2,≤_2,ω_2]]) be a matrix rings over skew generalized power series rings, where R_1,R_2 are commutative rings with an identity element, (S_1,≤_1 ),(S_2,≤_2 ) are strictly ordered monoids, ω_1:S_1→End(R_1 ),〖 ω〗_2:S_2→End(R_2 ) are monoid homomorphisms. In this research, a mapping  τ from M_n (R_1 [[S_1,≤_1,ω_1]]) to M_n (R_2 [[S_2,≤_2,ω_2]]) is defined by using a strictly ordered monoid homomorphism δ:(S_1,≤_1 )→(S_2,≤_2 ), and ring homomorphisms μ:R_1→R_2 and σ:R_1 [[S_1,≤_1,ω_1]]→R_2 [[S_2,≤_2,ω_2]]. Furthermore, it is proved that τ is a ring homomorphism, and also the sufficient conditions for  τ to be a monomorphism, epimorphism, and isomorphism are given.

TRIANGULAR MATRIX REPRESENTATION OF SKEW GENERALIZED POWER SERIES RINGS

2012 ◽  
Vol 05 (04) ◽  
pp. 1250027 ◽  
Author(s):  
Amit Bhooshan Singh
Keyword(s):  
Power Series ◽  
Matrix Representation ◽  
Triangular Matrix ◽  
Group Rings ◽  
Power Series Ring ◽  
Generalized Power Series ◽  
Series Ring ◽  
Skew Polynomial Rings ◽  
Monoid Homomorphism ◽  
Power Series Rings

Let R be a ring, (S, ≤) a strictly ordered monoid and ω : S → End (R) a monoid homomorphism. In this paper, we study the triangular matrix representation of skew generalized power series ring R[[S, ω]] which is a compact generalization of (skew) polynomial rings, (skew) power series rings, (skew) Laurent polynomials rings, (skew) Laurent power series rings, (skew) group rings, (skew) monoid rings, Mal'cev–Neumann rings and generalized power series rings. We investigate that if R is S-compatible and (S, ω)-Armendariz, then the skew generalized power series ring has same triangulating dimension as R. Furthermore, if R is a PWP ring, then skew generalized power series is also PWP ring.

McCoy property and nilpotent elements of skew generalized power series rings

2017 ◽  
Vol 16 (10) ◽  
pp. 1750183 ◽  
Author(s):  
Kamal Paykan ◽  
Ahmad Moussavi
Keyword(s):  
Power Series ◽  
Sufficient Conditions ◽  
Polynomial Rings ◽  
Group Rings ◽  
Generalized Power Series ◽  
Series Ring ◽  
Skew Polynomial Rings ◽  
Monoid Homomorphism ◽  
Power Series Rings ◽  
Skew Polynomial

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 consider the problem of determining when [Formula: see text] is nilpotent in [Formula: see text]. We study various annihilator properties and a variety of conditions and related properties that the skew generalized power series [Formula: see text] inherits from [Formula: see text]. We also introduce and study the [Formula: see text]-McCoy condition on [Formula: see text], a generalization of the standard McCoy condition from polynomials to skew generalized power series. We resolve the structure of [Formula: see text]-McCoy rings and obtain various necessary or sufficient conditions for a ring to be [Formula: see text]-McCoy. As particular cases of our general results we obtain several new theorems on the McCoy condition. Moreover various examples of [Formula: see text]-McCoy rings are provided.

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