Commutative subsemigroups of the composition semigroup of formal power series over an integral domain

1979 ◽  
Vol 27 (3) ◽  
pp. 313-318
Author(s):  
Hermann Kautschitsch
Keyword(s):  
Power Series ◽  
Commutative Ring ◽  
Formal Power Series ◽  
Integral Domain ◽  
Positive Order ◽  
Formal Power

AbstractLet R be a commutative ring with identity. R[[x]] denotes the ring of formal power series, in which we consider the composition ○, defined by f(x)○g(x)=f(g(x)). This operation is well defined in the subring R+[[x]] of formal power series of positive order. The algebra= 〈R+[[x]], ○〉 is learly a semigroup, which is not commutative for ∣R∣>1. In this paper we consider all those commutative subsemigroups of , which consist of power series of all positive orders, which are called ‘permutable chains’.

scholarly journals The exponent of certain finite p-groups

1990 ◽  
Vol 33 (3) ◽  
pp. 483-490 ◽  
Author(s):  
I. O. York
Keyword(s):  
Power Series ◽  
Commutative Ring ◽  
Formal Power Series ◽  
Group Operation ◽  
Image Position ◽  
Formal Power ◽  
Quotient Groups

In this paper, for R a commutative ring, with identity, of characteristic p, we look at the group G(R) of formal power series with coefficients in R, of the formand the group operation being substitution. The results obtained give the exponent of the quotient groups Gn(R) of this group, n∈ℕ.

On S-GCD domains

2019 ◽  
Vol 18 (04) ◽  
pp. 1950067 ◽  
Author(s):  
D. D. Anderson ◽  
Ahmed Hamed ◽  
Muhammad Zafrullah
Keyword(s):  
Power Series ◽  
Formal Power Series ◽  
Integral Domain ◽  
Class Group ◽  
Nonzero Ideal ◽  
Noetherian Domain ◽  
Power Series Rings ◽  
Formal Power ◽  
Equivalent Conditions ◽  
Unit Formula

Let [Formula: see text] be a multiplicative set in an integral domain [Formula: see text]. A nonzero ideal [Formula: see text] of [Formula: see text] is said to be [Formula: see text]-[Formula: see text]-principal if there exist an [Formula: see text] and [Formula: see text] such that [Formula: see text]. Call [Formula: see text] an [Formula: see text]-GCD domain if each finitely generated nonzero ideal of [Formula: see text] is [Formula: see text]-[Formula: see text]-principal. This notion was introduced in [A. Hamed and S. Hizem, On the class group and [Formula: see text]-class group of formal power series rings, J. Pure Appl. Algebra 221 (2017) 2869–2879]. One aim of this paper is to characterize [Formula: see text]-GCD domains, giving several equivalent conditions and showing that if [Formula: see text] is an [Formula: see text]-GCD domain then [Formula: see text] is a GCD domain but not conversely. Also we prove that if [Formula: see text] is an [Formula: see text]-GCD [Formula: see text]-Noetherian domain such that every prime [Formula: see text]-ideal disjoint from [Formula: see text] is a [Formula: see text]-ideal, then [Formula: see text] is [Formula: see text]-factorial and we give an example of an [Formula: see text]-GCD [Formula: see text]-Noetherian domain which is not [Formula: see text]-factorial. We also consider polynomial and power series extensions of [Formula: see text]-GCD domains. We call [Formula: see text] a sublocally [Formula: see text]-GCD domain if [Formula: see text] is a [Formula: see text]-GCD domain for every non-unit [Formula: see text] and show, among other things, that a non-quasilocal sublocally [Formula: see text]-GCD domain is a generalized GCD domain (i.e. for all [Formula: see text] is invertible).

Algebras of Acyclic Type

1981 ◽  
Vol 33 (1) ◽  
pp. 129-141 ◽  
Author(s):  
Phil Hanlon
Keyword(s):  
Power Series ◽  
Commutative Ring ◽  
Formal Power Series ◽  
Homomorphic Image ◽  
Incidence Algebra ◽  
Section 8 ◽  
Slight Generalization ◽  
Background Material ◽  
Binomial Type ◽  
Formal Power

In this paper we consider the problem of determining when an algebra of formal power series over a commutative ring R is the homomorphic image of a reduced incidence algebra P(R, ∽). The question of when two such algebras are isomorphic is answered in Section 8 of [1]. A slight generalization of their notion of full binomial type is introduced here.Section 1 contains background material together writh a summary of the results of [1]. In Section 2 we present the desired characterization, and to conclude an application appears in Section 3. In Section 3 the tools of Section 2 are used to derive an equation of R. W. Robinson and R. P. Stanley which counts labelled, acyclic digraphs.

scholarly journals On F-integrable actions of the restricted Lie algebra of a formal group F in characteristic p > 0

1989 ◽  
Vol 115 ◽  
pp. 125-137 ◽  
Author(s):  
Andrzej Tyc
Keyword(s):  
Power Series ◽  
Lie Algebra ◽  
Lie Algebras ◽  
Formal Power Series ◽  
Integral Domain ◽  
Formal Group ◽  
Characteristic P ◽  
Restricted Lie Algebra ◽  
Formal Power

Let k be an integral domain, let F = (F1X, Y),…, Fn(X, Y)), X = (X1,…, Xn), Y = (Y1,…, Yn), be an n-dimensional formal group over k, and let L(F) be the Lie algebra of all F-invariant k-derivations of the ring of formal power series k[X] (cf. § 2). If A is a (commutative) k-algebra and Derk (A) denotes the Lie algebra of all k-derivations d: A → A, then by an action of L(F) on A we mean a morphism of Lie algebras φ: L(F) → Derk (A) such that φ(dp) = φ(d)p, provided char (k) = p > 0. An action of the formal group F on A is a morphism of k-algebras D: A-→A[X] such that D(a)≡a mod (X) for a ∊ A, and FA ∘ D = DY ∘ D, where FA: A[X] → A[X, Y], DF: A[X] → A[X, Y] are morphisms of k-algebras given by FA(g(X)) = g(F), DY(Σa aaXa) = Σa D(aa)Y; for a motivation of this notion, see [15]. Let D: A → A[X] be such an action.

Substitution Groups of Formal Power Series

1954 ◽  
Vol 6 ◽  
pp. 325-340 ◽  
Author(s):  
S. A. Jennings
Keyword(s):  
Power Series ◽  
Commutative Ring ◽  
Formal Power Series ◽  
Point Of View ◽  
Group Operation ◽  
Several Variables ◽  
Special Cases ◽  
Image Position ◽  
Formal Power ◽  
Complex Coefficients

In this paper we are concerned with the group of formal power series of the form,the coefficients being elements of a commutative ring R and the group operation being substitution. Little seems to be known of the properties of groups of this type, except in special cases, although groups of formal power series in several variables with complex coefficients have been investigated from a different point of view by Bochner and Martin (1, chap. I) and Gotô (2).

On transfer of annihilator conditions of rings

2018 ◽  
Vol 17 (10) ◽  
pp. 1850199
Author(s):  
Abdollah Alhevaz ◽  
Ebrahim Hashemi ◽  
Rasul Mohammadi
Keyword(s):  
Power Series ◽  
Commutative Ring ◽  
Formal Power Series ◽  
Noetherian Rings ◽  
Power Series Ring ◽  
Series Ring ◽  
Formal Power Series Ring ◽  
Base Ring ◽  
Polynomial Formula ◽  
Formal Power

It is well known that a polynomial [Formula: see text] over a commutative ring [Formula: see text] with identity is a zero-divisor in [Formula: see text] if and only if [Formula: see text] has a non-zero annihilator in the base ring, where [Formula: see text] is the polynomial ring with indeterminate [Formula: see text] over [Formula: see text]. But this result fails in non-commutative rings and in the case of formal power series ring. In this paper, we consider the problem of determining some annihilator properties of the formal power series ring [Formula: see text] over an associative non-commutative ring [Formula: see text]. We investigate relations between power series-wise McCoy property and other standard ring-theoretic properties. In this context, we consider right zip rings, right strongly [Formula: see text] rings and rings with right Property [Formula: see text]. We give a generalization (in the case of non-commutative ring) of a classical results related to the annihilator of formal power series rings over the commutative Noetherian rings. We also give a partial answer, in the case of formal power series ring, to the question posed in [1 Question, p. 16].

Nonnil-Noetherian Rings and Formal Power Series

2020 ◽  
Vol 27 (03) ◽  
pp. 361-368
Author(s):  
Ali Benhissi
Keyword(s):  
Power Series ◽  
Commutative Ring ◽  
Formal Power Series ◽  
Quotient Ring ◽  
Noetherian Rings ◽  
Power Series Ring ◽  
Series Ring ◽  
Formal Power

Let A be a commutative ring with unit. We characterize when A is nonnil-Noetherian in terms of the quotient ring A/ Nil(A) and in terms of the power series ring A[[X]].

scholarly journals NON-COMMUTATIVE CHARACTERISTIC POLYNOMIALS AND COHN LOCALIZATION

2001 ◽  
Vol 64 (1) ◽  
pp. 13-28 ◽  
Author(s):  
DESMOND SHEIHAM
Keyword(s):  
Power Series ◽  
Commutative Ring ◽  
Formal Power Series ◽  
Equivalence Class ◽  
Characteristic Polynomial ◽  
Constant Coefficient ◽  
Characteristic Polynomials ◽  
Invertible Matrix ◽  
Finitely Generated ◽  
Formal Power

Almkvist proved that for a commutative ring A the characteristic polynomial of an endomorphism α : P → P of a finitely generated projective A-module determines (P, α) up to extensions. For a non-commutative ring A the generalized characteristic polynomial of an endomorphism of an endomorphism α : P → P of a finitely generated projective A-module is defined to be the Whitehead torsion [1 − xα] ∈ K1(A[[x]]), which is an equivalence class of formal power series with constant coefficient 1.The paper gives an example of a non-commutative ring A and an endomorphism α : P → P for which the generalized characteristic polynomial does not determine (P, α) up to extensions. The phenomenon is traced back to the non-injectivity of the natural map [sum ]−1A[x] → A[[x]] where [sum ]−1A[x] is the Cohn localization of A[x] inverting the set [sum ] of matrices in A[x] sent to an invertible matrix by A[x] → A;x [map ] 0.

scholarly journals On the radius and the relation between the total graph of a commutative ring and its extensions

2011 ◽  
Vol 89 (103) ◽  
pp. 1-9 ◽  
Author(s):  
Zoran Pucanovic ◽  
Zoran Petrovic
Keyword(s):  
Power Series ◽  
Commutative Ring ◽  
Formal Power Series ◽  
Polynomial Rings ◽  
Total Graph ◽  
Formal Power

We discuss the determination of the radius of the total graph of a commutative ring R in the case when this graph is connected. Typical extensions such as polynomial rings, formal power series, idealization of the R-module M and relations between the total graph of the ring R and its extensions are also dealt with.

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