Unifying strongly clean power series rings

It is unknown whether a power series ring over a strongly clean ring is, itself, always strongly clean. Although a number of authors have shown that the above statement is true in certain special cases, the problem remains open, in general. In this article, we look at a class of strongly clean rings, which we call the optimally clean rings, over which power series are strongly clean. This condition is motivated by work in [10 Diesl, A. J., Dorsey, T. J., Garg, S., Khurana, D. (2012). A note on completeness and strongly clean rings, preprint. [Google Scholar]] and [11 Diesl, A. J., Dorsey, T. J., Iberkleid, W., LaFuente-Rodriguez, R., McGovern, W (2013). Strongly clean triangular matrices over abelian rings, preprint. [Google Scholar]]. We explore the properties of optimally clean rings and provide many examples, highlighting the role that this new class of rings plays in investigating the question of strongly clean power series.     

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.     

Composition rings from formal power series rings

We study the ideals in some Dickson nearrings of polynomials and formal power series. For some of their related quotients, we introduce variants and generalizations, and construct composition rings also.     

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.     

On power series rings over valuation domains

Let V be a valuation domain and V[[X]] be the power series ring over V. In this paper, we show that if V[[X]] is a locally finite intersection of valuation domains, then V is an SFT domain and hence a discrete valuation domain. As a consequence, it is shown that the power series ring V[[X]] is a Krull domain if and only if V[[X]] is a generalized Krull domain if and only if V[[X]] is an integral domain of Krull type (or equivalently, a PvMD of finite t-character) if and only if V is a discrete valuation domain with Krull dimension at most one.     

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.

     

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.     

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.     

Power series over strongly Hopfian bounded rings

Let R be a commutative ring with identity. We show that R[[X]] is strongly Hopfian bounded if and only if R has a strongly Hopfian bounded extension T such that I_c(T) contains a regular element of T. We deduce that if R[[X]] is strongly Hopfian bounded, then so is R[[X,Y]] where X,Y are two indeterminates over R. Also we show that if R is embeddable in a zero-dimensional strongly Hopfian bounded ring, then so is R[[X]] (this generalizes most results of Hizem [11 Hizem, S. (2011). Formal power series over strongly Hopfian rings. Commun. Algebra 39(1):279–291.[Taylor & Francis Online], [Web of Science ®] , [Google Scholar]]). For a chained ring R, we show that R[[X]] is strongly Hopfian if and only if R is strongly Hopfian.     

Unique factorization in generalized power series rings

Let be a field of characteristic zero and let denote the ring of generalized power series (i.e., formal sums with well-ordered support) with coefficients in , and non-positive real exponents. Berarducci (2000) constructed an irreducible omnific integer, in the sense of Conway (2001), by first proving that an element of that is not divisible by a monomial and whose support has order type (or for some ordinal ) must be irreducible. In this paper, we consider elements of with support of order type . The irreducibility of these elements cannot be deduced solely from the order type of their support and, after developing new tools for studying these elements, we exhibit both reducible and irreducible elements of this type. We further prove that all elements whose support has order type and which are not divisible by a monomial factor uniquely into irreducibles. This provides, in the ring , a class of reducible elements for which we have unique factorization into irreducibles.

     

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.     

Noetherian-like properties in polynomial and power series rings

There are many Noetherian-like rings. Among them, we are interested in SFT-rings, piecewise Noetherian rings, and rings with Noetherian prime spectrum. Some of them are stable under polynomial extensions but none of them are stable under power series extensions. We give partial answers to some open questions related with stabilities of such rings. In particular, we show that any mixed extensions $R[X_1??[X_n?$ over a zero-dimensional SFT ring R are also SFT-rings, and that if R is an SFT-domain such that $R/P$ is integrally closed for each prime ideal P of R, then $R[X]$ is an SFT-ring. We also give a direct proof that if R is an SFT Prüfer domain, then $R[X_1,?,X_n]$ is an SFT-ring. Finally, we show that the power series extension $R?X?$ over a Prüfer domain R is piecewise Noetherian if and only if R is Noetherian.     

Exchange rings satisfying power ideal-cancellation

In this paper, we introduce a new kind of partial power cancellations, and get a number of necessary and sufficient conditions for exchange rings satisfying power ideal-cancellation. These also extend many known results.     

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.