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.

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.

scholarly journals A fresh look into monoid rings and formal power series rings

2019 ◽  
Vol 19 (01) ◽  
pp. 2050003
Author(s):  
Abolfazl Tarizadeh
Keyword(s):  
Power Series ◽  
Formal Power Series ◽  
Universal Property ◽  
Polynomial Rings ◽  
Power Series Rings ◽  
Universal Properties ◽  
Ring Of Polynomials ◽  
Formal Power

In this paper, the ring of polynomials is studied in a systematic way through the theory of monoid rings. As a consequence, this study provides canonical approaches in order to find easy and rigorous proofs and methods for many facts on polynomials and formal power series; some of them as sample are treated in this paper. Besides the universal properties of the monoid rings and polynomial rings, a universal property for the formal power series rings is also established.

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.

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’.

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].

Annihilator Conditions on Ore Extensions

2008 ◽  
Vol 15 (02) ◽  
pp. 293-302
Author(s):  
E. Hashemi
Keyword(s):  
Power Series ◽  
Formal Power Series ◽  
Polynomial Rings ◽  
Skew Polynomials ◽  
Baer Ring ◽  
Skew Polynomial Rings ◽  
Ore Extensions ◽  
Skew Polynomial ◽  
Armendariz Ring ◽  
Formal Power

In this paper, we extend the study of various annihilator conditions on the nearring of polynomials and the nearring of formal power series to skew polynomials and skew formal power series in which addition and substitution are used as operations. A result of Hong et al. on the skew polynomial rings and skew power series of an α-rigid Baer ring is extended to Baer conditions in the nearrings of skew polynomials and skew formal power series. Also, for an injective endomorphism α of a ring R, it is shown that R is α-rigid if and only if the nearring (R[x; α], +, ◦) is reduced, if and only if the nearring (R0[[x; α]], +, ◦) is reduced, if and only if R is a reduced and near Armendariz ring.

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.

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