Hurwitz Series Ring Research Articles

Hurwitz series rings satisfying a zero divisor property

In this paper, we study zero divisors in the Hurwitz series rings and the Hurwitz polynomial rings over general non-commutative rings. We first construct Armendariz rings that are not Armendariz of the Hurwitz series type and find various properties of (the Hurwitz series) Armendariz rings. We show that for a semiprime Armendariz of the Hurwitz series type (so reduced) ring [Formula: see text] with [Formula: see text] on annihilator ideals, HR (the Hurwitz series ring with coefficients over [Formula: see text]) has finitely many minimal prime ideals, say [Formula: see text] such that [Formula: see text] and [Formula: see text] for some minimal prime ideal [Formula: see text] of [Formula: see text] for all [Formula: see text], where [Formula: see text] are all minimal prime ideals of [Formula: see text]. Additionally, we construct various types of (the Hurwitz series) Armendariz rings and demonstrate that the polynomial ring extension preserves the Armendarizness of the Hurwitz series as the Armendarizness.

Weakly p.q.-Baer skew Hurwitz series rings

A ring [Formula: see text] is projective invariant Baer ([Formula: see text]-Baer for short) if the right annihilator of every projection invariant left ideal of [Formula: see text] is generated by an idempotent element of [Formula: see text]. A ring [Formula: see text] is called weakly principally quasi-Baer or simply (weakly p.q.-Baer) if the right annihilator of a principal right ideal is left [Formula: see text]-unital by left semicentral idempotents, which implies that [Formula: see text] modulo, the right annihilator of any principal right ideal, is flat. We study relations between the [Formula: see text]-Baer and weakly p.q.-Baer properties of a ring [Formula: see text], and its skew Hurwitz series ring [Formula: see text], where [Formula: see text] is a ring equipped with an endomorphism [Formula: see text]. Examples are provided to explain the results.

Almost strong finite type rings and Krull dimension of power series ring extensions from sequences

Let [Formula: see text] be the collection of power series with coefficients in a commutative ring [Formula: see text] with identity. For a suitable function [Formula: see text] from [Formula: see text] to [Formula: see text], one can define a multiplication [Formula: see text] in [Formula: see text] such that together with the usual addition, [Formula: see text] becomes a ring that contains [Formula: see text] as a subring. Denote this ring by [Formula: see text]. By this observation, the usual power series ring and the well-known Hurwitz series ring are the special cases of [Formula: see text] when [Formula: see text] for all [Formula: see text] and [Formula: see text] for all [Formula: see text], respectively. In this paper, we study the Krull dimension of [Formula: see text]. We first introduce the concept of almost strong finite type (ASFT) rings and study basic properties of this type of rings. We then show that for any function [Formula: see text], the Krull dimension of [Formula: see text] is at least [Formula: see text] if [Formula: see text] is a non-ASFT ring, which is an analogue of the result about the Krull dimension of the usual power series ring that [Formula: see text] if [Formula: see text] is a non-SFT ring. In particular, we have the Krull dimension of the Hurwitz series ring is at least [Formula: see text] if [Formula: see text] is a non-ASFT ring, which gives an answer to a question of Benhissi and Koja about the infiniteness of the Krull dimension of the Hurwitz series ring.

Duo property for rings by the quasinilpotent perspective

In this paper, we focus on the duo ring property via quasinilpotent elements, which gives a new kind of generalizations of commutativity. We call this kind of rings qnil-duo. Firstly, some properties of quasinilpotents in a ring are provided. Then the set of quasinilpotents is applied to the duo property of rings, in this perspective, we introduce and study right (resp., left) qnil-duo rings. We show that this concept is not left-right symmetric. Among others, it is proved that if the Hurwitz series ring $H(R; \alpha)$ is right qnil-duo, then $R$ is right qnil-duo. Every right qnil-duo ring is abelian. A right qnil-duo exchange ring has stable range 1.

Reversibility of skew Hurwitz series rings

We study the reversibility of skew Hurwitz series at zero as a generalization of an $\alpha$-rigid ring, introducing the concept of skew Hurwitz reversibility. A ring $R$ is called skew Hurwitz reversible ($SH$-reversible, for short), if the skew Hurwitz series ring $(HR,\alpha)$ is reversible i.e. whenever skew Hurwitz series $f, g\in (HR,\alpha)$ satisfy $fg=0$, then $gf=0$. We examine some characterizations and extensions of $SH$-reversible rings in relation with several ring theoretic properties which have roles in ring theory.

Skew Hurwitz serieswise quasi-Armendariz rings

For a ring endomorphism [Formula: see text], a generalization of semiprime rings and right p.q.-Baer rings, which we call quasi-Armendariz rings of skew Hurwitz series type (or simply, [Formula: see text]-[Formula: see text]), is introduced and studied. It is shown that the [Formula: see text]-rings are closed upper triangular matrix rings, full matrix rings and Morita invariance. Some characterizations for the skew Hurwitz series ring [Formula: see text] to be quasi-Baer, generalized quasi-Baer, primary, nilary, reflexive, ideal-symmetric and semiprime are concluded.

McCoy property of Hurwitz series rings

Based on a theorem of McCoy on commutative rings, Nielsen called a ring [Formula: see text] right McCoy if for any nonzero polynomials [Formula: see text] over [Formula: see text], [Formula: see text] implies [Formula: see text] for some [Formula: see text]. In this note, we introduce and investigate McCoy and [Formula: see text]-properties of Hurwitz series ring [Formula: see text] and its Hurwitz polynomial subring [Formula: see text]. We show that when [Formula: see text] is a reversible or duo ring and [Formula: see text] then the Hurwitz polynomial ring [Formula: see text] is McCoy.

On a conjecture posed by Benhissi and Koja

Let = (Rn)n≥0 be an ascending chain of commutative rings with identity and (resp., ) the generalized composite Hurwitz series ring (resp., generalized composite Hurwitz polynomial ring). In this paper, we completely characterize when the rings and are PS-rings. More precisely, we show that and are always PS-rings. This is an affirmative answer to the question raised by Benhissi and Koja.Communicated by Silvana Bazzoni

Composite Hurwitz Rings as PF-Rings and PP-Rings

Let R ⊆ T be an extension of commutative rings with identity and H ( R , T ) (respectively, h ( R , T ) ) the composite Hurwitz series ring (respectively, composite Hurwitz polynomial ring). In this article, we study equivalent conditions for the rings H ( R , T ) and h ( R , T ) to be PF-rings and PP-rings. We also give some examples of PP-rings and PF-rings via the rings H ( R , T ) and h ( R , T ) .

Further results on skew Hurwitz series ring (I)

In this paper, we continue the study of skew Hurwitz series ring $$(H R, \alpha )$$ , where R is a ring equipped with an endomorphism $$\alpha $$ . In particular, we investigate the problem when a skew Hurwitz series series ring $$(HR, \alpha )$$ 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 Hurwitz series rings to be right non-singular.

Special properties of Hurwitz series rings

In this paper, we study some properties of the Hurwitz series ring HR (resp. Hurwitz polynomial ring hR), such as the flatness or the faithful flatness of HR / (f) (resp. hR / (f)), the strongly Hopfian property and the radical property of HR (resp. hR). We give some sufficient and necessary conditions for HR / (f) (resp. hR / (f)) to be flat or faithful flat. We also prove that the strongly Hopfian property transfer between R and HR (resp. hR), and some radicals of HR can be determined in terms of those of R, in case R satisfies some additional conditions.

Some results on skew Hurwitz series rings

A ring is quasi-Baer (respectively, Baer) in case the right annihilator of every ideal (respectively, subset) is generated by an idempotent, as a right ideal. In this article, we investigate on the relationship between the quasi-Baerness, Baerness, semiprimitivity, simplicity and NI properties of a ring R, and its skew Hurwitz series ring $$(HR, \alpha )$$ , where R is a ring equipped with an endomorphism $$\alpha $$ .

On the operations of sequences in rings and binomial type sequences

Given a commutative ring with identity $R$, many different and interesting operations can be defined over the set $H_R$ of sequences of elements in $R$. These operations can also give $H_R$ the structure of a ring. We study some of these operations, focusing on the binomial convolution product and the operation induced by the composition of exponential generating functions. We provide new relations between these operations and their invertible elements. We also study automorphisms of the Hurwitz series ring, highlighting that some well--known transforms of sequences (such as the Stirling transform) are special cases of these automorphisms. Moreover, we introduce a novel isomorphism between $H_R$ equipped with the componentwise sum and the set of the sequences starting with 1 equipped with the binomial convolution product. Finally, thanks to this isomorphism, we find a new method for characterizing and generating all the binomial type sequences.

Properties of Hurwitz polynomial and Hurwitz series rings

Properties of Hurwitz polynomial and Hurwitz series rings

Chain conditions on composite Hurwitz series rings

Abstract In this paper, we study chain conditions on composite Hurwitz series rings and composite Hurwitz polynomial rings. More precisely, we characterize when composite Hurwitz series rings and composite Hurwitz polynomial rings are Noetherian, S-Noetherian or satisfy the ascending chain condition on principal ideals.

Some combinatorial properties of the Hurwitz series ring

We study some properties and perspectives of the Hurwitz series ring $$H_R[[t]]$$ , for an integral domain R, with multiplicative identity and zero characteristic. Specifically, we provide a closed form for the invertible elements by means of the complete ordinary Bell polynomials, we highlight some connections with well–known transforms of sequences, and we see that the Stirling transforms are automorphisms of $$H_R[[t]]$$ . Moreover, we focus the attention on some special subgroups studying their properties. Finally, we introduce a new transform of sequences that allows to see one of this subgroup as an ultrametric dynamic space.

A study on skew Hurwitz series rings

In this paper, we continue the study of skew Hurwitz series ring \((HR, \alpha )\), where R is a ring equipped with an endomorphism \(\alpha \). Necessary and sufficient conditions are obtained for \((HR, \alpha )\) to satisfy a certain ring property which is among being local, semilocal, semiperfect, left quasi-duo, clean, exchange, right stable range one, 2-good, projective-free, semiprime, semiregular, I-ring, respectively. Furthermore, we prove that \((HR,\alpha )\) is a domain satisfying the ascending chain condition on principal left (resp. right) ideals if and only if R is a domain satisfying the ascending chain condition on principal left (resp. right) ideals, \(\alpha \) is injective and \(char(R)=0\). Finally, we study the prime radical of skew Hurwitz series ring.

Principally quasi-Baer skew Hurwitz series rings

A ring is quasi-Baer (respectively, right p.q.-Baer) in case the right annihilator of every (respectively, principal right) ideal is generated by an idempotent, as a right ideal. A ring R is right AIP if the right annihilator of any right ideal of R is pure as a right ideal. In this article, we study relations between the quasi-Baer, right p.q.-Baer, and right AIP properties of a ring R, and its skew Hurwitz series ring \((HR, \alpha )\), where R is a ring equipped with an endomorphism \(\alpha \). Examples to illustrate and delimit the theory are provided.

Nilpotent elements of skew Hurwitz series rings

For a ring endomorphism \(\alpha \), we consider the problem of determining when an element f of the skew Hurwitz series ring \( (HR, \alpha )\) is nilpotent. As an application, it is shown that the skew Hurwitz series ring \( (HR, \alpha )\) is weak zip (resp. weak symmetric) if and only if R is weak zip (resp. weak symmetric) under some additional conditions. We also characterize when a skew Hurwitz series ring is 2-primal.

Chain Condition on Annihilators and Strongly Hopfian Property in Hurwitz Series Ring

In this paper we study some properties of the Hurwitz series ring HA over a commutative ring A, such as the nilradical of HA and the chain condition on its annihilators. We provide an example showing that the last property does not pass from A to HA. A strongly Hopfian ring is a ring satisfying the chain condition on some type of annihilators. We give a large class of strongly Hopfian rings A such that HA are not strongly Hopfian.