Semiprimeness,quasi-Baerness and prime radical of skew generalized power series rings

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) power series rings, (skew) Laurent polynomial rings, (skew) group rings, and Mal’cev-Neumann Laurent series rings. In this paper we obtain necessary and sufficient conditions for the skew generalized power series ring R[[S,ω]] to be a semiprime, prime, quasi-Baer, or Baer ring. Furthermore, we study the prime radical of a skew generalized power series ring R[[S,ω]]. Our results extend and unify many existing results. In particular, we obtain new theorems on (skew) group rings, Mal’cev-Neumann Laurent series rings and the ring of generalized power series.     

Uniserial rings of skew generalized power series

Let R be a ring, S a strictly ordered monoid and a monoid homomorphism. In this paper we obtain some necessary conditions for the skew generalized power series ring RS,ω to be right (respectively left) uniserial, and we prove that these conditions are also sufficient when the monoid S is commutative or totally ordered.     

Special Properties of Rings of Skew Generalized Power Series

Let R be a ring, S a strictly ordered monoid, and ω: S → End(R) a monoid homomorphism. In [30 Marks , G. , Mazurek , R. , Ziembowski , M. ( 2010 ). A unified approach to various generalizations of Armendariz rings . Bull. Aust. Math. Soc. 81 : 361 – 397 .[Crossref], [Web of Science ®] , [Google Scholar]], Marks, Mazurek, and Ziembowski study the (S, ω)-Armendariz condition on R, a generalization of the standard Armendariz condition from polynomials to skew generalized power series. Following [30 Marks , G. , Mazurek , R. , Ziembowski , M. ( 2010 ). A unified approach to various generalizations of Armendariz rings . Bull. Aust. Math. Soc. 81 : 361 – 397 .[Crossref], [Web of Science ®] , [Google Scholar]], we provide various classes of nonreduced (S, ω)-Armendariz rings, and determine radicals of the skew generalized power series ring R[[S ^≤, ω]], in terms of those of an (S, ω)-Armendariz ring R. We also obtain some characterizations for a skew generalized power series ring to be local, semilocal, clean, exchange, uniquely clean, 2-primal, or symmetric.     

Some minimal rings related to 2-primal rings

In a paper on the taxonomy of 2-primal rings, examples of various types of rings that are related to commutativity such as reduced, symmetric, duo, reversible and PS I were given in order to show that the ring class inclusions were strict. Many of the rings given in the examples were infinite. In this paper, where possible, examples of minimal finite rings of the various types are given. Along with the rings in the previous taxonomy, NI, abelian and reflexive rings are also included.     

Descriptions of radicals of skew polynomial and skew Laurent polynomial rings

In this paper, we first characterize the Levitzki radical of a skew (Laurent) polynomial ring by the prime ideals and skewed prime ideals in the base ring. We next provide formulas for the strongly prime radical and the uniformly strongly prime radical of skew (Laurent) polynomial rings.     

Krull property of generalized power series rings

Let D be an integral domain and let $(S,\le )$ be a torsion-free, ≤-cancellative, subtotally ordered monoid. We show that the generalized power series ring $?D^{S,\le }?$ is a Krull domain if and only if D is a Krull domain and S is a Krull monoid.     

On inverse skew Laurent series extensions of weakly rigid rings

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.     

On the integral dependence of power series rings

We prove the following results: (1) Let R ? S be two commutative rings. Suppose that dim R = 0.If f(X) ∈ S[[X]]is integral over R[[X]], then every coefficient of f(X) is integral over R. (2) Let dim R ≥ 1. There exists a ring S containing R and a power series f(X) ∈ S[[X]]such that f(X) is integral over R[[X]], but not all coefficients of f(X) are integral over R. (3) Let k ? R. Suppose that R is algebraic over the field k. Then R[[X]] is integral over k[[X]] if and only if the nilradical of R is nilpotent and the separable degree and the inseparable exponent of R _red over k are finite.     

Principal quasi-Baerness of formal power series rings

Let R be a ring. We consider left （or right） principal quasi-Baerness of the left skew formal power series ring R[[x;α]] over R where a is a ring automorphism of R. We give a necessary and sufficient condition under which the ring R[[x; α]] is left （or right） principally quasi-Baer. As an application we show that R[[x]] is left principally quasi-Baer if and only if R is left principally quasi- Baer and the left annihilator of the left ideal generated by any countable family of idempotents in R is generated by an idempotent.     

Commutator automorphisms of formal power series rings

For a big class of commutative rings , every continuous -automorphism of with the linear part the identity is in the commutator subgroup of . An explicit bound for the number of commutators involved and a -theoretic interpretation of this result are provided.

     

On extensions of matrix rings with skew Hochschild 2-cocycles

We study structures of Hochschild 2-cocycles related to endomorphisms and introduce a skew Hochschild 2-cocycle. We moreover define skew Hochschild extensions equipped with skew Hochschild 2-cocycles, and then we examine uniquely clean, Abelian, directly finite, symmetric, and reversible ring properties of skew Hochschild extensions and related ring systems. The results obtained here provide various kinds of examples of such rings. Especially, we give an answer negatively to the question of H. Lin and C. Xi for the corresponding Hochschild extensions of reversible (or semicommutative) rings. Finally, we establish three kinds of Hochschild extensions with Hochschild 2-cocycles and skew Hochschild 2-cocycles.     

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

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) power series rings, (skew) Laurent polynomial rings, (skew) group rings, and Mal'cev–Neumann Laurent series rings. In this article, we study relations between the (quasi-) Baer, principally quasi-Baer and principally projective properties of a ring R, and its skew generalized power series extension R[[S, ω]]. As particular cases of our general results, we obtain new theorems on (skew) group rings, Mal'cev–Neumann Laurent series rings, and the ring of generalized power series.     

Computation of generalized real radicals of polynomial ideals

For an ordered field (K,T) and an idealI of the polynomial ring , the construction of the generalized real radical ofI is investigated. When (K,T) satisfies some computational requirements, a method of computing is presented. Project supported by the National Natural Science Foundation of China (Grant No. 19661002) and the Climbing Project.     

Some Triple Block Intersection Numbers of Paley 2-Designs of QN-type

We determine here some possible values for thecardinality of the intersection of three blocks from Paley2-(2q+1, q, (q-1)/2) designs where qis a prime power such that (mod 4).