Generalized Power Series Rings Research Articles

Semi-Baer and Semi-Quasi Baer Properties of Skew Generalized Power Series Rings

Semi-Baer and Semi-Quasi Baer Properties of Skew Generalized Power Series Rings

Regularity of twisted partial skew generalized power series

In this article, we work with a unital twisted partial action of a group G on a ring R. We study when the twisted partial skew generalized power series rings are Von Newman regular. Moreover, we study when the partial skew Malcev-Newman ring is simple, we provide some examples where the assumption of our principal results about Von Newman regularity can not be dropped and we discuss a conjecture given by Tuganbaev in the case of partial actions.

Root extension of generalized power series rings

Let R 1 ⊆ R 2 be commutative rings with unity and let S 1 ⊆ S 2 be torsion-free abelian monoids. We give a characterization of when the extension R 1 [ S 1 ] ⊆ R 2 [ S 2 ] of monoid rings is a root extension. For torsion-free, ≤ -cancellative, strictly ordered abelian monoids ( S 1 , ≤ ) ⊆ ( S 2 , ≤ ) , we also give a characterization of when the extension [ ​ [ R 1 S 1 , ≤ ] ​ ] ⊆ [ ​ [ R 2 S 2 , ≤ ] ​ ] of generalized power series rings is a root extension.

Krull property of generalized power series rings

Let D be an integral domain and let ( S , ≤ ) be a torsion-free, ≤-cancellative, subtotally ordered monoid. We show that the generalized power series ring 〚 D S , ≤ 〛 is a Krull domain if and only if D is a Krull domain and S is a Krull monoid.

Some Ring Structures of Skew Genaralized Power Series Rings

In the current article, we are going to investigate a relationship between some abstract ring structures (weakly PP-rings, clean rings, uniquely clean rings and n-clean rings) along with the skew generalized power series rings A[[N, ϑ]], where A is one of the ring structures described above, (N, ≤) represents a strictly ordered monoid while ϑ : N → End(A) represents a monoid homomorphism. We shall propose unified extensions of the above-mentioned ring structures by employing specific conditions along with their proofs.

On the P.Q.-Baer Skew Generalized Power Series Modules

For a ring [Formula: see text] and a strictly totally ordered monoid [Formula: see text], let [Formula: see text] be a monoid homomorphism and [Formula: see text] an [Formula: see text]-weakly rigid right [Formula: see text]-module (i.e., for any elements [Formula: see text], [Formula: see text] and [Formula: see text], [Formula: see text] if and only if [Formula: see text]), where [Formula: see text] is the ring of ring endomorphisms of [Formula: see text]. It is shown that the skew generalized power series module [Formula: see text] is a principally quasi-Baer module if and only if the annihilator of every submodule generated by an [Formula: see text]-indexed subset of [Formula: see text] is generated by an idempotent as a right ideal of [Formula: see text]. As a consequence we deduce that for an [Formula: see text]-weakly rigid ring [Formula: see text], the skew generalized power series ring [Formula: see text] is right principally quasi-Baer if and only if [Formula: see text] is right principally quasi-Baer and any [Formula: see text]-indexed subset of right semicentral idempotents in [Formula: see text] has a generalized [Formula: see text]-indexed join in [Formula: see text]. The range of previous results in this area is expanded by these results.

IDEMPOTENT MATRIX OVER SKEW GENERALIZED POWER SERIES RINGS

Let $R[[S,\leq,\omega]]$ be a skew generalized power series ring, with $R$ is a ring with an identity element, $(S,\leq)$ a strictly ordered monoid, and $\omega:S\rightarrow End(R)$ a monoid homomorphism. We define the set of all matrices over $R[[S,\leq,\omega]]$, denoted by $M_{n}(R[[S,\leq,\omega]])$. With the addition and multiplication matrix operations, $M_{n}(R[[S,\leq,\omega]])$ becomes a ring. In this paper, we determine the sufficient conditions for $R$, $(S,\leq)$, and $\omega$, so the element of $M_{n}(R[[S,\leq,\omega]])$ is an idempotent matrix.

Zero-divisor graphs of twisted partial skew generalized power series rings

PurposeThe aim of this paper is to investigate the relationship between the ring structure of the twisted partial skew generalized power series ring RG,≤;Θ and the corresponding structure of its zero-divisor graph Γ̅RG,≤;Θ.Design/methodology/approachThe authors first introduce the history and motivation of this paper. Secondly, the authors give a brief exposition of twisted partial skew generalized power series ring, in addition to presenting some properties of such structure, for instance, a-rigid ring, a-compatible ring and (G,a)-McCoy ring. Finally, the study’s main results are stated and proved.FindingsThe authors establish the relation between the diameter and girth of the zero-divisor graph of twisted partial skew generalized power series ring RG,≤;Θ and the zero-divisor graph of the ground ring R. The authors also provide counterexamples to demonstrate that some conditions of the results are not redundant. As well the authors indicate that some conditions of recent results can be omitted.Originality/valueThe results of the twisted partial skew generalized power series ring embrace a wide range of results of classical ring theoretic extensions, including Laurent (skew Laurent) polynomial ring, Laurent (skew Laurent) power series ring and group (skew group) ring and of course their partial skew versions.

The Ring Homomorphisms of Matrix Rings over Skew Generalized 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.

Weakly principally quasi-Baer skew generalized power series rings

Let $$(S,\le )$$ be a strictly totally ordered monoid and R an $$(S,\omega )$$ -weakly rigid ring, where $$\omega :S\rightarrow End(R)$$ is a monoid homomorphism. In this paper, we study the weakly p.q.-Bear property of the skew generalized power series ring $$R[[S,\omega ]]$$ . As a consequence, the weakly p.q.-Baer property of the skew power series ring $$R[[x;\alpha ]]$$ and the skew Laurent power series ring $$R[[x,x^{-1};\alpha ]]$$ are determined, where $$\alpha$$ is a ring endomorphism of R.

MATRIKS ATAS RING DERET PANGKAT TERGENERALISASI MIRING

Let R be a ring with unit elements, strictly ordered monoids, and a monoid homomorphism. Formed , which is a set of all functions from S to R with are Artin and narrow. With the operation of the sum of functions and convolution multiplication, is a ring, from now on referred to as the Skew Generalized Power Series Ring (SGPSR). In this paper, the set of all matrices over SGPSR will be constructed. Furthermore, it will be shown that this set is a ring with the addition and multiplication matrix operations. Moreover, we will construct the ideal of ring matrix over SGPSR and investigate this ideal's properties.

Triangulating dimension of skew generalized power series rings

Let [Formula: see text] be a ring, [Formula: see text] a strictly ordered monoid and [Formula: see text] a monoid homomorphism. In this paper, we investigate the problem when a skew generalized power series ring [Formula: see text] has the same triangulating dimension as the ring [Formula: see text]. Furthermore, for a piecewise prime ring we determine a large class of the skew generalized power series ring which have a generalized triangular matrix representation for which the diagonal rings are prime. Our results extend and unify many existing results.

Noetherian properties in composite generalized power series rings

AbstractLet(Γ,≤)({\mathrm{\Gamma}},\le )be a strictly ordered monoid, and letΓ⁎=Γ\{0}{{\mathrm{\Gamma}}}^{\ast }\left={\mathrm{\Gamma}}\backslash \{0\}. LetD⊆ED\subseteq Ebe an extension of commutative rings with identity, and letIbe a nonzero proper ideal ofD. SetD+〚EΓ⁎,≤〛≔f∈〚EΓ,≤〛|f(0)∈DandD+〚IΓ⁎,≤〛≔f∈〚DΓ,≤〛|f(α)∈I,forallα∈Γ⁎.\begin{array}{l}D+[\kern-2pt[ {E}^{{{\mathrm{\Gamma}}}^{\ast },\le }]\kern-2pt] := \left\{f\in [\kern-2pt[ {E}^{{\mathrm{\Gamma}},\le }]\kern-2pt] \hspace{0.15em}|\hspace{0.2em}f(0)\in D\right\}\hspace{.5em}\text{and}\\ \hspace{0.2em}D+[\kern-2pt[ {I}^{{\Gamma }^{\ast },\le }]\kern-2pt] := \left\{f\in [\kern-2pt[ {D}^{{\mathrm{\Gamma}},\le }]\kern-2pt] \hspace{0.15em}|\hspace{0.2em}f(\alpha )\in I,\hspace{.5em}\text{for}\hspace{.25em}\text{all}\hspace{.5em}\alpha \in {{\mathrm{\Gamma}}}^{\ast }\right\}.\end{array}In this paper, we give necessary conditions for the ringsD+〚EΓ⁎,≤〛D+[\kern-2pt[ {E}^{{{\mathrm{\Gamma}}}^{\ast },\le }]\kern-2pt]to be Noetherian when(Γ,≤)({\mathrm{\Gamma}},\le )is positively ordered, and sufficient conditions for the ringsD+〚EΓ⁎,≤〛D+[\kern-2pt[ {E}^{{{\mathrm{\Gamma}}}^{\ast },\le }]\kern-2pt]to be Noetherian when(Γ,≤)({\mathrm{\Gamma}},\le )is positively totally ordered. Moreover, we give a necessary and sufficient condition for the ringD+〚IΓ⁎,≤〛D+[\kern-2pt[ {I}^{{\Gamma }^{\ast },\le }]\kern-2pt]to be Noetherian when(Γ,≤)({\mathrm{\Gamma}},\le )is positively totally ordered. As corollaries, we give equivalent conditions for the ringsD+(X1,…,Xn)E[X1,…,Xn]D+({X}_{1},\ldots ,{X}_{n})E{[}{X}_{1},\ldots ,{X}_{n}]andD+(X1,…,Xn)I[X1,…,Xn]D+({X}_{1},\ldots ,{X}_{n})I{[}{X}_{1},\ldots ,{X}_{n}]to be Noetherian.

The X[[S]]-Sub-Exact Sequence of Generalized Power Series Rings

Let be a ring, a strictly ordered monoid, and K, L, M are R-modules. Then, we can construct the Generalized Power Series Modules (GPSM) K[[S]], L[[S]], and M[[S]], which are the module over the Generalized Power Series Rings (GPSR) R[[S]]. In this paper, we investigate the property of X[[S]]-sub-exact sequence on GPSM L[[S]] over GPSR R[[S]].

Goldie ranks of skew generalized power series rings

A skew generalized power series ring consists of all functions from a strictly ordered monoid S to a ring R whose support contains neither infinite descending chains nor infinite antichains, 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 the skew generalized power series 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 “untwisted” versions of all of these, and generalized power series rings. In this paper, we continue the study of skew generalized power series ring We investigate the problem when a skew generalized power series ring has the same Goldie rank as the ring R, and we obtain partial characterizations for it to be serial semiprime. Finally, we will obtain criterion for skew generalized power series rings to be right nonsingular.

Some characterizations of 2-primal skew generalized power series rings

A skew generalized power series ring consists of all functions from a strictly ordered monoid S to a ring R whose support contains neither infinite descending chains nor infinite antichains, 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 the skew generalized power series 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 “untwisted” versions of all of these, and generalized power series rings. In this paper we obtain necessary and sufficient conditions on R, S, and ω such that the skew generalized power series ring is a 2-primal ring. In particular, it is proved that, under suitable conditions, is 2-primal if and only if for every minimal prime ideal of there exists a minimal prime ideal P of R such that P is completely prime and Examples to illustrate and delimit the theory are provided.Communicated by Jose Luis Gomez Pardo

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

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.

The ascending chain condition on principal ideals in composite generalized power series rings

Let $D \subseteq E$ be an extension of commutative rings with identity, $I$ a nonzero proper ideal of $D$, $(\Gamma , \leq )$ a strictly totally ordered monoid such that $0 \leq \alpha $ for all $\alpha \in \Gamma $, and $\Gamma ^*=\Gamma \setminus \{0\}$. Let $D+[\![E^{\Gamma ^*, \leq }]\!]=\{f \in [\![E^{\Gamma , \leq }]\!] \mid f(0) \in D\}$ and $D+[\![I^{\Gamma ^*, \leq }]\!] =\{f \in [\![D^{\Gamma , \leq }]\!] \mid f(\alpha ) \in I$ for all $\alpha \in \Gamma ^*\}$. In this paper, we give some conditions for the rings $D+[\![E^{\Gamma ^*, \leq }]\!]$ and $D+[\![I^{\Gamma ^*, \leq }]\!]$ to satisfy the ascending chain condition on principal ideals.

Skew Generalized Power Series Rings and the McCoy Property

Given a ring $R$, a strictly totally ordered monoid $(S,\preceq)$ and a monoid homomorphism $\omega \colon S \to \operatorname{End}(R)$, one can construct the skew generalized power series ring $R[[S,\omega,\preceq]]$, consisting all of the functions from a monoid $S$ to a coefficient ring $R$ whose support is artinian and narrow, where the addition is pointwise, and the multiplication is given by convolution twisted by an action $\omega$ of the monoid $S$ on the ring $R$. In this paper, we consider the problem of determining some annihilator and zero-divisor properties of the skew generalized power series ring $R[[S,\omega,\preceq]]$ over an associative non-commutative ring $R$. Providing many examples, we investigate relations between McCoy property of skew generalized power series ring, namely $(S,\omega)$-McCoy property, and other standard ring-theoretic properties. We show that if $R$ is a local ring such that its Jacobson radical $J(R)$ is nilpotent, then $R$ is $(S,\omega)$-McCoy. Also if $R$ is a semicommutative semiregular ring such that $J(R)$ is nilpotent, then $R$ is $(S,\omega)$-McCoy ring.

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

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].