scholarly journals On separable cyclic extensions of rings

1982 ◽  
Vol 32 (2) ◽  
pp. 165-170 ◽  
Author(s):  
George Szeto ◽  
Yuen-Fat Wong
Keyword(s):  
Commutative Ring ◽  
Quaternion Algebra ◽  
Cyclic Extension ◽  
Noncommutative Ring ◽  
Azumaya Algebras ◽  
Galois Extensions

AbstractThe quaternion algebra of degree 2 over a commutative ring as defined by S. Parimala and R. Sridharan is generalized to a separable cyclic extension B[j] of degree n over a noncommutative ring B. A characterization of such an extension is given, and a relation between Azumaya algebras and Galois extensions for B[j] is also obtained.

A characterization of projective weak Galois extensions

2009 ◽  
Vol 174 (1) ◽  
pp. 161-177 ◽  
Author(s):  
J. N. Alonso Álvarez ◽  
J. M. Fernández Vilaboa ◽  
R. González Rodríguez ◽  
M. P. López
Keyword(s):  
Galois Extensions

Complete Decomposability in the Exterior Algebra of a Free Module

1980 ◽  
Vol 32 (1) ◽  
pp. 27-33 ◽  
Author(s):  
M. Boratynski ◽  
E. D. Davis ◽  
A. V. Geramita
Keyword(s):  
Commutative Ring ◽  
Present Note ◽  
Free Module ◽  
Exterior Algebra ◽  
Standard Bases ◽  
Classical Criterion ◽  
Generally True

Recall the classical criterion for the complete decomposability of exterior vectors: the completely decomposable vectors in ∧pRn, R a field, are precisely the “Plücker vectors,” i.e. those whose coordinates (relative to the standard bases for ∧pRn) satisfy the Plücker equations. For R an arbitrary commutative ring, completely decomposable exterior vectors are still Plücker vectors, but the converse is not generally true. Rings for which the converse is true (for all 1 ≤ p ≤ n) are called Towber rings. Noetherian Towber rings are regular and, in fact, are characterized by the property that every Plücker vector in ∧2R4 is completely decomposable. (See [10] for these two results as well as for the above mentioned facts.) The present note develops a new characterization of Towber rings, combining it with results of Kleiner [9] and Estes-Matijevic [5] in (1) below.

CLEAN, ALMOST CLEAN, POTENT COMMUTATIVE RINGS

2007 ◽  
Vol 06 (04) ◽  
pp. 671-685 ◽  
Author(s):  
K. VARADARAJAN
Keyword(s):  
Commutative Ring ◽  
Analogous Result ◽  
Commutative Rings ◽  
Complete Characterization ◽  
Zariski Topology ◽  
Clean Ring ◽  
Regular Rings

We give a complete characterization of the class of commutative rings R possessing the property that Spec(R) is weakly 0-dimensional. They turn out to be the same as strongly π-regular rings. We considerably strengthen the results of K. Samei [13] tying up cleanness of R with the zero dimensionality of Max(R) in the Zariski topology. In the class of rings C(X), W. Wm Mc Govern [6] has characterized potent rings as the ones with X admitting a clopen π-base. We prove the analogous result for any commutative ring in terms of the Zariski topology on Max(R). Mc Govern also introduced the concept of an almost clean ring and proved that C(X) is almost clean if and only if it is clean. We prove a similar result for all Gelfand rings R with J(R) = 0.

scholarly journals A characterization of minimal prime ideals

1998 ◽  
Vol 40 (2) ◽  
pp. 223-236 ◽  
Author(s):  
Gary F. Birkenmeier ◽  
Jin Yong Kim ◽  
Jae Keol Park
Keyword(s):  
Positive Integer ◽  
Prime Ideal ◽  
Commutative Ring ◽  
Minimal Prime Ideal ◽  
Prime Ideals ◽  
Sheaf Representations ◽  
Minimal Prime

AbstractLet P be a prime ideal of a ring R, O(P) = {a ∊ R | aRs = 0, for some s ∊ R/P} | and Ō(P) = {x ∊ R | xn ∊ O(P), for some positive integer n}. Several authors have obtained sheaf representations of rings whose stalks are of the form R/O(P). Also in a commutative ring a minimal prime ideal has been characterized as a prime ideal P such that P= Ō(P). In this paper we derive various conditions which ensure that a prime ideal P = Ō(P). The property that P = Ō(P) is then used to obtain conditions which determine when R/O(P) has a unique minimal prime ideal. Various generalizations of O(P) and Ō(P) are considered. Examples are provided to illustrate and delimit our results.

scholarly journals A Characterization of separable polynomials over a skew polynomial ring

1985 ◽  
Vol 38 (2) ◽  
pp. 275-280
Author(s):  
George Szeto
Keyword(s):  
Commutative Ring ◽  
Finite Order ◽  
Polynomial Ring ◽  
Subject Classification ◽  
Skew Polynomial Ring ◽  
Skew Polynomial

AbstractThe characterization of a separable polynomial over an indecomposable commutative ring (with no idempotents but 0 and 1) in terms of the discriminant proved by G. J. Janusz is generalized to a skew polynomial ring R [ X, ρ] over a not necessarily commutative ring R where ρ is an automorphism of R with a finite order. 1980 Mathematics subject classification (Amer. Math. Soc.): 16 A 05.

scholarly journals Purely infinite simple Kumjian–Pask algebras

2018 ◽  
Vol 30 (1) ◽  
pp. 253-268 ◽  
Author(s):  
Hossein Larki
Keyword(s):  
Commutative Ring ◽  
Path Algebras ◽  
Higher Rank ◽  
Leavitt Path Algebras ◽  
Unital Simple

Abstract Given any finitely aligned higher-rank graph Λ and any unital commutative ring R, the Kumjian–Pask algebra {\mathrm{KP}_{R}(\Lambda)} is known as the higher-rank generalization of Leavitt path algebras. After the characterization of simple Kumjian–Pask algebras by Clark and Pangalela among others, in this article we focus on the purely infinite simple ones. Briefly, we show that if {\mathrm{KP}_{R}(\Lambda)} is simple and every vertex of Λ is reached from a generalized cycle with an entrance, then {\mathrm{KP}_{R}(\Lambda)} is purely infinite. We also prove a dichotomy for simple Kumjian–Pask algebras: If each vertex of Λ is reached only from finitely many vertices and {\mathrm{KP}_{R}(\Lambda)} is simple, then {\mathrm{KP}_{R}(\Lambda)} is either purely infinite or locally matritial. This result covers all unital simple Kumjian–Pask algebras.

Rings with A Finitely Generated Total Quotient Ring

1974 ◽  
Vol 17 (1) ◽  
pp. 1-4 ◽  
Author(s):  
John Conway Adams
Keyword(s):  
Prime Ideal ◽  
Commutative Ring ◽  
Quotient Ring ◽  
Finitely Generated ◽  
Zero Divisors ◽  
Rank One ◽  
Definition Of

Let R be a commutative ring with non-zero identity and let K be the total quotient ring of R. We call R a G-ring if K is finitely generated as a ring over R. This generalizes Kaplansky′s definition of G-domain [5].Let Z(R) be the set of zero divisors in R. Following [7] elements of R—Z(R) and ideals of R containing at least one such element are called regular. Artin-Tate's characterization of Noetherian G-domains [1, Theorem 4] carries over with a slight adjustment to characterize a Noetherian G-ring as being semi-local in which every regular prime ideal has rank one.

scholarly journals On generalized quaternion algebras

1980 ◽  
Vol 3 (2) ◽  
pp. 237-245 ◽  
Author(s):  
George Szeto
Keyword(s):  
Automorphism Group ◽  
Commutative Ring ◽  
Galois Extension ◽  
Ring Extension ◽  
Separable Extension ◽  
Quaternion Algebras ◽  
Generalized Quaternion

LetBbe a commutative ring with1, andG(={σ})an automorphism group ofBof order2. The generalized quaternion ring extensionB[j]overBis defined byS. Parimala andR. Sridharan such that (1)B[j]is a freeB-module with a basis{1,j}, and (2)j2=−1andjb=σ(b)jfor eachbinB. The purpose of this paper is to study the separability ofB[j]. The separable extension ofB[j]overBis characterized in terms of the trace(=1+σ)ofBover the subring of fixed elements underσ. Also, the characterization of a Galois extension of a commutative ring given by Parimala and Sridharan is improved.

scholarly journals Noncommutative motives of Azumaya algebras

2014 ◽  
Vol 14 (2) ◽  
pp. 379-403 ◽  
Author(s):  
Gonçalo Tabuada ◽  
Michel Van den Bergh
Keyword(s):  
Commutative Ring ◽  
Global Dimension ◽  
Azumaya Algebras ◽  
Finite Global Dimension ◽  
Finite Dimensional ◽  
Noncommutative Motives ◽  
Finite Dimensional Algebras ◽  
The Way

AbstractLet $k$ be a base commutative ring, $R$ a commutative ring of coefficients, $X$ a quasi-compact quasi-separated $k$-scheme, and $A$ a sheaf of Azumaya algebras over $X$ of rank $r$. Under the assumption that $1/r\in R$, we prove that the noncommutative motives with $R$-coefficients of $X$ and $A$ are isomorphic. As an application, we conclude that a similar isomorphism holds for every $R$-linear additive invariant. This leads to several computations. Along the way we show that, in the case of finite-dimensional algebras of finite global dimension, all additive invariants are nilinvariant.

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