scholarly journals Composite Hurwitz Rings as PF-Rings and PP-Rings

Mathematics ◽  
2020 ◽  
Vol 8 (1) ◽  
pp. 100
Author(s):  
Dong Kyu Kim ◽  
Jung Wook Lim
Keyword(s):  
Polynomial Ring ◽  
Commutative Rings ◽  
Series Ring ◽  
Hurwitz Polynomial ◽  
Equivalent Conditions

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 ) .

McCoy property of Hurwitz series rings

2020 ◽  
pp. 2150105
Author(s):  
Vahid Nourozi ◽  
Farhad Rahmati ◽  
Morteza Ahmadi
Keyword(s):  
Polynomial Ring ◽  
Commutative Rings ◽  
Series Ring ◽  
Hurwitz Polynomial

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.

scholarly journals Noetherian properties in composite generalized power series rings

2020 ◽  
Vol 18 (1) ◽  
pp. 1540-1551
Author(s):  
Jung Wook Lim ◽  
Dong Yeol Oh
Keyword(s):  
Necessary Conditions ◽  
Sufficient Conditions ◽  
Commutative Rings ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Generalized Power Series ◽  
Ordered Monoid ◽  
Power Series Rings ◽  
Necessary And Sufficient ◽  
Equivalent Conditions

Abstract Let ({\mathrm{\Gamma}},\le ) be a strictly ordered monoid, and let {{\mathrm{\Gamma}}}^{\ast }\left={\mathrm{\Gamma}}\backslash \{0\} . Let D\subseteq E be an extension of commutative rings with identity, and let I be a nonzero proper ideal of D. Set \begin{array}{l}D+[\kern-2pt[ {E}^{{{\mathrm{\Gamma}}}^{\ast },\le }]\kern-2pt] := \left\{f\in [\kern-2pt[ {E}^{{\mathrm{\Gamma}},\le }]\kern-2pt] \hspace{0.15em}|\hspace{0.2em}f(0)\in D\right\}\hspace{.5em}\text{and}\\ \hspace{0.2em}D+[\kern-2pt[ {I}^{{\Gamma }^{\ast },\le }]\kern-2pt] := \left\{f\in [\kern-2pt[ {D}^{{\mathrm{\Gamma}},\le }]\kern-2pt] \hspace{0.15em}|\hspace{0.2em}f(\alpha )\in I,\hspace{.5em}\text{for}\hspace{.25em}\text{all}\hspace{.5em}\alpha \in {{\mathrm{\Gamma}}}^{\ast }\right\}.\end{array} In this paper, we give necessary conditions for the rings D+[\kern-2pt[ {E}^{{{\mathrm{\Gamma}}}^{\ast },\le }]\kern-2pt] to be Noetherian when ({\mathrm{\Gamma}},\le ) is positively ordered, and sufficient conditions for the rings D+[\kern-2pt[ {E}^{{{\mathrm{\Gamma}}}^{\ast },\le }]\kern-2pt] to be Noetherian when ({\mathrm{\Gamma}},\le ) is positively totally ordered. Moreover, we give a necessary and sufficient condition for the ring D+[\kern-2pt[ {I}^{{\Gamma }^{\ast },\le }]\kern-2pt] to be Noetherian when ({\mathrm{\Gamma}},\le ) is positively totally ordered. As corollaries, we give equivalent conditions for the rings D+({X}_{1},\ldots ,{X}_{n})E{[}{X}_{1},\ldots ,{X}_{n}] and D+({X}_{1},\ldots ,{X}_{n})I{[}{X}_{1},\ldots ,{X}_{n}] to be Noetherian.

scholarly journals Unique factorization in polynomial rings with zero divisors

2020 ◽  
pp. 2150113 ◽  
Author(s):  
D. D. Anderson ◽  
Ranthony A. C. Edmonds
Keyword(s):  
Commutative Ring ◽  
Polynomial Ring ◽  
Commutative Rings ◽  
Polynomial Rings ◽  
Unique Factorization ◽  
Factorization Property ◽  
Zero Divisors ◽  
Unique Factorization Domain

Given a certain factorization property of a ring [Formula: see text], we can ask if this property extends to the polynomial ring over [Formula: see text] or vice versa. For example, it is well known that [Formula: see text] is a unique factorization domain if and only if [Formula: see text] is a unique factorization domain. If [Formula: see text] is not a domain, this is no longer true. In this paper, we survey unique factorization in commutative rings with zero divisors, and characterize when a polynomial ring over an arbitrary commutative ring has unique factorization.

scholarly journals Singular ideals of skew Hurwitz polynomial rings

2018 ◽  
Vol 68 (3) ◽  
pp. 589-593
Author(s):  
Morteza Ahmadi
Keyword(s):  
Polynomial Ring ◽  
Polynomial Rings ◽  
Full Description ◽  
Hurwitz Polynomial ◽  
Left And Right

Abstract For a ring R and an endomorphism $$\alpha $$ α of R, we provide a full description of left and right singular ideals of the skew Hurwitz polynomial ring $$(hR,\alpha )$$ ( h R , α ) . We obtain that if $$\alpha $$ α is an automorphism of R, then R is right (resp., left) nonsingular if and only if $$ (hR,\alpha ) $$ ( h R , α ) is right (resp., left) nonsingular. We give an example of a ring R and an endomorphism $$\alpha $$ α of R such that the skew Hurwitz polynomial ring $$(hR,\alpha )$$ ( h R , α ) is left nonsingular, but not right nonsingular.

McCOY PROPERTY OF SKEW LAURENT POLYNOMIALS AND POWER SERIES RINGS

2013 ◽  
Vol 13 (02) ◽  
pp. 1350083 ◽  
Author(s):  
A. ALHEVAZ ◽  
D. KIANI
Keyword(s):  
Power Series ◽  
Commutative Ring ◽  
Commutative Rings ◽  
Laurent Polynomials ◽  
Property A ◽  
Power Series Ring ◽  
Series Ring ◽  
Monoid Ring ◽  
Power Series Rings ◽  
Base Ring

One of the important properties of commutative rings, proved by McCoy [Remarks on divisors of zero, Amer. Math. Monthly49(5) (1942) 286–295], is that if two nonzero polynomials annihilate each other over a commutative ring then each polynomial has a nonzero annihilator in the base ring. Nielsen [Semi-commutativity and the McCoy condition, J. Algebra298(1) (2006) 134–141] generalizes this property to non-commutative rings. Let M be a monoid and σ be an automorphism of a ring R. For the continuation of McCoy property of non-commutative rings, in this paper, we extend the McCoy's theorem to skew Laurent power series ring R[[x, x-1; σ]] and skew monoid ring R * M over general non-commutative rings. Constructing various examples, we classify how these skew versions of McCoy property behaves under various ring extensions. Moreover, we investigate relations between these properties and other standard ring-theoretic properties such as zip rings and rings with Property (A). As a consequence we extend and unify several known results related to McCoy rings.

Weakly Regular Rings

1973 ◽  
Vol 16 (3) ◽  
pp. 317-321 ◽  
Author(s):  
V. S. Ramamurthi
Keyword(s):  
Left Ideal ◽  
Commutative Rings ◽  
Weak Regularity ◽  
Left And Right ◽  
Equivalent Conditions ◽  
Regular Rings

This paper attempts to generalize a property of regular rings, namely,I2=I for every right (left) ideal. Rings with this property are called right (left) weakly regular. A ring which is both left and right weakly regular is called weakly regular. Kovacs in [6] proved that, for commutative rings, weak regularity and regularity are equivalent conditions and left open the question whether for arbitrary rings the two conditions are equivalent. We show in §1 that, in general weak regularity does not imply regularity. In fact, the class of weakly regular rings strictly contains the class of regular rings as well as the class of biregular rings.

scholarly journals Nilpotents and units in skew polynomial rings over commutative rings

1979 ◽  
Vol 28 (4) ◽  
pp. 423-426 ◽  
Author(s):  
M. Rimmer ◽  
K. R. Pearson
Keyword(s):  
Commutative Ring ◽  
Finite Order ◽  
Polynomial Ring ◽  
Commutative Rings ◽  
Polynomial Rings ◽  
Skew Polynomial Ring ◽  
Nilpotent Elements ◽  
Skew Polynomial Rings ◽  
Skew Polynomial

AbstractLet R be a commutative ring with an automorphism ∞ of finite order n. An element f of the skew polynomial ring R[x, α] is nilpotent if and only if all coefficients of fn are nilpotent. (The case n = 1 is the well-known description of the nilpotent elements of the ordinary polynomial ring R[x].) A characterization of the units in R[x, α] is also given.

Insertion of units at zero products

2018 ◽  
Vol 17 (03) ◽  
pp. 1850043 ◽  
Author(s):  
Hong Kee Kim ◽  
Tai Keun Kwak ◽  
Yang Lee ◽  
Yeonsook Seo
Keyword(s):  
Polynomial Ring ◽  
Jacobson Radical ◽  
Ring Theory ◽  
Noncommutative Ring ◽  
Matrix Rings ◽  
Zero Divisors ◽  
Noncommutative Ring Theory ◽  
Equivalent Conditions

The purpose of this paper is to provide useful connections between units and zero divisors, by investigating the structure of a class of rings in which Köthe’s conjecture (i.e. the sum of two nil left ideals is nil) holds. We introduce the concept of unit-IFP for the purpose, in relation with the inserting property of units at zero products. We first study the relation between unit-IFP rings and related ring properties in a kind of matrix rings which has roles in noncommutative ring theory. The Jacobson radical of the polynomial ring over a unit-IFP ring is shown to be nil. We also provide equivalent conditions to the commutativity via the unit-IFP of such matrix rings. We construct examples and counterexamples which are necessary to the naturally raised questions.

CONSTRUCTING CHAINS OF PRIMES IN POWER SERIES RINGS, II

2012 ◽  
Vol 12 (01) ◽  
pp. 1250123 ◽  
Author(s):  
K. ALAN LOPER ◽  
THOMAS G. LUCAS
Keyword(s):  
Lower Bound ◽  
Power Series ◽  
Polynomial Ring ◽  
Integral Domain ◽  
Valuation Domain ◽  
Main Concern ◽  
Power Series Ring ◽  
One Dimensional ◽  
Series Ring ◽  
Power Series Rings

For an integral domain D of dimension n, the dimension of the polynomial ring D[ x ] is known to be bounded by n + 1 and 2n + 1. While n + 1 is a lower bound for the dimension of the power series ring D[[ x ]], it often happens that D[[ x ]] has infinite chains of primes. For example, such chains exist if D is either an almost Dedekind domain that is not Dedekind or a one-dimensional nondiscrete valuation domain. The main concern here is in developing a scheme by which such chains can be constructed in the gap between MV[[ x ]] and M[[ x ]] when V is a one-dimensional nondiscrete valuation domain with maximal ideal M. A consequence of these constructions is that there are chains of primes similar to the set of ω1 transfinite sequences of 0's and 1's ordered lexicographically.

An intermediate ring between a polynomial ring and a power series ring

2013 ◽  
Vol 130 (1) ◽  
pp. 1-17
Author(s):  
M. Tamer Koşan ◽  
Tsiu-Kwen Lee ◽  
Yiqiang Zhou
Keyword(s):  
Power Series ◽  
Polynomial Ring ◽  
Power Series Ring ◽  
Series Ring ◽  
Intermediate Ring
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