scholarly journals Standard bases in mixed power series and polynomial rings over rings

2017 ◽  
Vol 79 ◽  
pp. 119-139 ◽  
Author(s):  
Thomas Markwig ◽  
Yue Ren ◽  
Oliver Wienand
Keyword(s):  
Power Series ◽  
Polynomial Rings ◽  
Standard Bases ◽  
Mixed Power

scholarly journals Generic fiber rings of mixed power series/polynomial rings

2006 ◽  
Vol 298 (1) ◽  
pp. 248-272 ◽  
Author(s):  
William Heinzer ◽  
Christel Rotthaus ◽  
Sylvia Wiegand
Keyword(s):  
Power Series ◽  
Polynomial Rings ◽  
Generic Fiber ◽  
Mixed Power

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.

On semilocal, Bézout and distributive generalized power series rings

2015 ◽  
Vol 25 (05) ◽  
pp. 725-744 ◽  
Author(s):  
Ryszard Mazurek ◽  
Michał Ziembowski
Keyword(s):  
Power Series ◽  
Neumann Series ◽  
Polynomial Rings ◽  
Group Rings ◽  
Power Series Ring ◽  
Generalized Power Series ◽  
Series Ring ◽  
Ordered Monoid ◽  
Power Series Rings ◽  
Totally Ordered Monoid

Let R be a ring, and let S be a strictly ordered monoid. The generalized power series ring R[[S]] is a common generalization of polynomial rings, Laurent polynomial rings, power series rings, Laurent series rings, Mal'cev–Neumann series rings, monoid rings and group rings. In this paper, we examine which conditions on R and S are necessary and which are sufficient for the generalized power series ring R[[S]] to be semilocal right Bézout or semilocal right distributive. In the case where S is a strictly totally ordered monoid we characterize generalized power series rings R[[S]] that are semilocal right distributive or semilocal right Bézout (the latter under the additional assumption that S is not a group).

scholarly journals Pole assignability in polynomial rings, power series rings, and Prüfer domains

1987 ◽  
Vol 106 (1) ◽  
pp. 265-286 ◽  
Author(s):  
James Brewer ◽  
Daniel Katz ◽  
William Ullery
Keyword(s):  
Power Series ◽  
Polynomial Rings ◽  
Power Series Rings ◽  
Prufer Domains ◽  
Prüfer Domains

scholarly journals STANDARD BASES OVER RINGS

2010 ◽  
Vol 20 (07) ◽  
pp. 953-968 ◽  
Author(s):  
AFSHAN SADIQ
Keyword(s):  
Noetherian Ring ◽  
Linear Equations ◽  
Polynomial Rings ◽  
Commutative Noetherian Ring ◽  
Standard Bases

The theory of standard bases in polynomial rings with coefficients in a ring R with respect to local orderings is developed. R is a commutative Noetherian ring with 1 and we assume that linear equations are solvable in R.

INTERPRETING ARITHMETIC IN THE FIRST-ORDER THEORY OF ADDITION AND COPRIMALITY OF POLYNOMIAL RINGS

2019 ◽  
Vol 84 (3) ◽  
pp. 1194-1214
Author(s):  
JAVIER UTRERAS
Keyword(s):  
Power Series ◽  
Entire Functions ◽  
Order Theory ◽  
Ring Structure ◽  
Polynomial Rings ◽  
Series Ring ◽  
First Order ◽  
First Order Theory ◽  
Power Series Rings ◽  
Image Position

AbstractWe study the first-order theory of polynomial rings over a GCD domain and of the ring of formal entire functions over a non-Archimedean field in the language $\{ 1, + , \bot \}$. We show that these structures interpret the first-order theory of the semi-ring of natural numbers. Moreover, this interpretation depends only on the characteristic of the original ring, and thus we obtain uniform undecidability results for these polynomial and entire functions rings of a fixed characteristic. This work enhances results of Raphael Robinson on essential undecidability of some polynomial or formal power series rings in languages that contain no symbols related to the polynomial or power series ring structure itself.

Nilpotent elements and nil-Armendariz property of skew generalized power series rings

2016 ◽  
Vol 10 (02) ◽  
pp. 1750034 ◽  
Author(s):  
Kamal Paykan ◽  
Ahmad Moussavi
Keyword(s):  
Power Series ◽  
Sufficient Conditions ◽  
Polynomial Rings ◽  
Group Rings ◽  
Power Series Ring ◽  
Generalized Power Series ◽  
Series Ring ◽  
Skew Polynomial Rings ◽  
Monoid Homomorphism ◽  
Power Series Rings

Let [Formula: see text] be a ring, [Formula: see text] a strictly ordered monoid, and [Formula: see text] a monoid homomorphism. The skew generalized power series ring [Formula: see text] 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 introduce and study the [Formula: see text]-nil-Armendariz condition on [Formula: see text], a generalization of the standard nil-Armendariz condition from polynomials to skew generalized power series. We resolve the structure of [Formula: see text]-nil-Armendariz rings and obtain various necessary or sufficient conditions for a ring to be [Formula: see text]-nil-Armendariz. The [Formula: see text]-nil-Armendariz condition is connected to the question of whether or not a skew generalized power series ring [Formula: see text] over a nil ring [Formula: see text] is nil, which is related to a question of Amitsur [Algebras over infinite fields, Proc. Amer. Math. Soc. 7 (1956) 35–48]. As particular cases of our general results we obtain several new theorems on the nil-Armendariz condition. Our results extend and unify many existing results.

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