scholarly journals A UNIFIED APPROACH TO VARIOUS GENERALIZATIONS OF ARMENDARIZ RINGS

2010 ◽  
Vol 81 (3) ◽  
pp. 361-397 ◽  
Author(s):  
GREG MARKS ◽  
RYSZARD MAZUREK ◽  
MICHAŁ ZIEMBOWSKI
Keyword(s):  
Power Series ◽  
Sufficient Conditions ◽  
Polynomial Rings ◽  
Power Series Ring ◽  
Generalized Power Series ◽  
Series Ring ◽  
Skew Polynomial Rings ◽  
Monoid Homomorphism ◽  
Special Cases ◽  
Armendariz Rings

AbstractLet 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. We study the (S,ω)-Armendariz condition on R, a generalization of the standard Armendariz condition from polynomials to skew generalized power series. We resolve the structure of (S,ω)-Armendariz rings and obtain various necessary or sufficient conditions for a ring to be (S,ω)-Armendariz, unifying and generalizing a number of known Armendariz-like conditions in the aforementioned special cases. As particular cases of our general results we obtain several new theorems on the Armendariz condition; for example, left uniserial rings are Armendariz. We also characterize when a skew generalized power series ring is reduced or semicommutative, and we obtain partial characterizations for it to be reversible or 2-primal.

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.

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 LEFT APP-RINGS OF SKEW GENERALIZED POWER SERIES

2011 ◽  
Vol 10 (05) ◽  
pp. 891-900 ◽  
Author(s):  
RENYU ZHAO
Keyword(s):  
Power Series ◽  
Chain Condition ◽  
Polynomial Rings ◽  
Group Rings ◽  
Power Series Ring ◽  
Generalized Power Series ◽  
Series Ring ◽  
Ordered Monoid ◽  
Skew Polynomial Rings ◽  
Monoid Homomorphism

A ring R is called a left APP-ring if the left annihilator lR(Ra) is right s-unital as an ideal of R for any a ∈ R. Let R be a ring, (S, ≤) be a commutative strictly ordered monoid and ω: S → End (R) be a monoid homomorphism. The skew generalized power series ring [[RS, ≤, ω]] is a common generalization of (skew) polynomial rings, (skew) power series rings, (skew) Laurent polynomial rings, (skew) group rings and Malcev–Neumann Laurent series rings. We study the left APP-property of the skew generalized power series ring [[RS, ≤, ω]]. It is shown that if (S, ≤) is a commutative strictly totally ordered monoid, ω: S→ Aut (R) a monoid homomorphism and R a ring satisfying the descending chain condition on right annihilators, then [[RS, ≤, ω]] is left APP if and only if for any S-indexed subset A of R, the ideal lR(∑a ∈ A ∑s ∈ S Rωs (a)) is right s-unital.

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.

A GENERALIZATION OF NIL-ARMENDARIZ RINGS

2013 ◽  
Vol 12 (06) ◽  
pp. 1350001 ◽  
Author(s):  
MOHAMMAD HABIBI ◽  
RAOUFEH MANAVIYAT
Keyword(s):  
Sufficient Conditions ◽  
Laurent Polynomial ◽  
Polynomial Rings ◽  
Monoid Ring ◽  
Skew Polynomial Rings ◽  
Monoid Homomorphism ◽  
Special Cases ◽  
Armendariz Rings ◽  
Skew Polynomial ◽  
The Relationship

Let R be a ring, M a monoid and ω : M → End (R) a monoid homomorphism. The skew monoid ring R * M is a common generalization of polynomial rings, skew polynomial rings, (skew) Laurent polynomial rings and monoid rings. In the current work, we study the nil skew M-Armendariz condition on R, a generalization of the standard nil-Armendariz condition from polynomials to skew monoid rings. We resolve the structure of nil skew M-Armendariz rings and obtain various necessary or sufficient conditions for a ring to be nil skew M-Armendariz, unifying and generalizing a number of known nil Armendariz-like conditions in the aforementioned special cases. We consider central idempotents which are invariant under a monoid endomorphism of nil skew M-Armendariz rings and classify how the nil skew M-Armendariz rings behaves under various ring extensions. We also provide rich classes of skew monoid rings which satisfy in a condition nil (R * M) = nil (R) * M. Moreover, we study on the relationship between the zip and weak zip properties of a ring R and those of the skew monoid ring R * M.

Quasi-Armendariz generalized power series rings

2016 ◽  
Vol 15 (05) ◽  
pp. 1650086 ◽  
Author(s):  
K. Paykan ◽  
A. Moussavi
Keyword(s):  
Power Series ◽  
Triangular Matrix ◽  
Polynomial Rings ◽  
Power Series Ring ◽  
Matrix Rings ◽  
Generalized Power Series ◽  
Series Ring ◽  
Armendariz Rings ◽  
Power Series Rings ◽  
Upper Triangular Matrix

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. We initiate the study of the [Formula: see text]-quasi-Armendariz condition on [Formula: see text], a generalization of the standard quasi-Armendariz condition from polynomials to skew generalized power series. The class of quasi-Armendariz rings includes semiprime rings, Armendariz rings, right (left) p.q.-Baer rings and right (left) PP rings. The [Formula: see text]-quasi-Armendariz rings are closed under direct sums, upper triangular matrix rings, full matrix rings and Morita invariance. The [Formula: see text] formal upper triangular matrix rings of this class are characterized. We conclude some characterizations for a skew generalized power series ring to be semiprime, quasi-Baer, generalized quasi-Baer, primary, nilary, reflexive, ideal-symmetric and left AIP. Examples to illustrate and delimit the theory are provided.

Archimedean domains of skew generalized power series

2020 ◽  
Vol 32 (4) ◽  
pp. 1075-1093
Author(s):  
Ryszard Mazurek
Keyword(s):  
Power Series ◽  
Neumann Series ◽  
Polynomial Rings ◽  
Group Rings ◽  
Generalized Power Series ◽  
Series Ring ◽  
Skew Polynomial Rings ◽  
Open Questions ◽  
Special Cases ◽  
Power Series Rings

AbstractA skew generalized power series ring {R[[S,\omega,\leq]]} consists of all functions from a strictly ordered monoid {(S,\leq)} to a ring R whose support is artinian and narrow, with pointwise addition, and with multiplication given by convolution twisted by an action ω of the monoid S on the ring R. Special cases of this ring construction are skew polynomial rings, skew Laurent polynomial rings, skew power series rings, skew Laurent series rings, skew monoid rings, skew group rings, skew Mal’cev–Neumann series rings, the “unskewed” versions of all of these, and generalized power series rings. In this paper, we characterize the skew generalized power series rings {R[[S,\omega,\leq]]} that are left (right) Archimedean domains in the case where the order {\leq} is total, or {\leq} is semisubtotal and the monoid S is commutative torsion-free cancellative, or {\leq} is trivial and S is totally orderable. We also answer four open questions posed by Moussavi, Padashnik and Paykan regarding the rings in the title.

scholarly journals The Sufficient Conditions for M[[S,w]] to be T[[S,w]]-Noetherian R[[S,w]]-module

2019 ◽  
Vol 10 (2) ◽  
pp. 285-292
Author(s):  
Ahmad Faisol ◽  
Fitriani Fitriani
Keyword(s):  
Power Series ◽  
Sufficient Conditions ◽  
Power Series Ring ◽  
Generalized Power Series ◽  
Series Ring ◽  
Ordered Monoid ◽  
Monoid Homomorphism

In this paper, we investigate the sufficient conditions for T[[S,w]] to be a multiplicative subset of skew generalized power series ring R[[S,w]], where R is a ring, T Í R a multiplicative set, (S,≤) a strictly ordered monoid, and w : S®End(R) a monoid homomorphism. Furthermore, we obtain sufficient conditions for skew generalized power series module M[[S,w]] to be a T[[S,w]]-Noetherian R[[S,w]]-module, where M is an R-module.

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