A theorem in Banach algebras and its applications

1979 ◽  
Vol 20 (2) ◽  
pp. 247-252 ◽  
Author(s):  
V.K. Srinivasan ◽  
Hu Shaing
Keyword(s):  
Banach Algebra ◽  
Principal Ideal ◽  
Banach Algebras ◽  
Negative Integer ◽  
Complex Field ◽  
Principal Ideal Domain ◽  
Bezout Domain ◽  
Ideal Domain ◽  
Complex Banach Algebra ◽  
Divisor Of Zero

If A is a complex Banach algebra which is also a Bezout domain, it is shown that for any prime p and a non-negative integer n, pn is not a topological divisor of zero. Using the above result it is shown that a complex Banach algebra which is a principal ideal domain is isomorphic to the complex field.

scholarly journals On some Gelfand-Mazur like theorems in Banach algebras

1979 ◽  
Vol 20 (2) ◽  
pp. 211-215 ◽  
Author(s):  
V.K. Srinivasan
Keyword(s):  
Banach Algebra ◽  
Integral Domain ◽  
Banach Algebras ◽  
Complex Field ◽  
Principal Ideal Domain ◽  
Locally Finite ◽  
Several Variables ◽  
Complex Banach Algebra ◽  
Noetherian Domain ◽  
Formal Power

The following Gelfand-Mazur like theorems are proved in this paper:(1) A complex Banach algebra which is locally finite, and which is also an integral domain, is isomorphic to the complex field .(2) A complex Banach algebra which is a noetherian domain is isomorphic to .(3) A complex Banach algebra which is a principal ideal domain is isomorphic to .An application is given to the algebra of all complex formal power series in several variables.

scholarly journals Notes on topological rings

2013 ◽  
Vol 29 (2) ◽  
pp. 267-273
Author(s):  
MIHAIL URSUL ◽  
◽  
MARTIN JURAS ◽  
Keyword(s):  
Principal Ideal ◽  
Principal Ideal Domain ◽  
Compact Ring ◽  
Ring Topology ◽  
Bezout Domain ◽  
Totally Bounded ◽  
Ideal Domain ◽  
Nilpotent Ring ◽  
Trivial Multiplication

We prove that every infinite nilpotent ring R admits a ring topology T for which (R, T ) has an open totally bounded countable subring with trivial multiplication. A new example of a compact ring R for which R2 is not closed, is given. We prove that every compact Bezout domain is a principal ideal domain.

DECIDABILITY OF MODULES OVER A BÉZOUT DOMAIN D+XQ[X] WITH D A PRINCIPAL IDEAL DOMAIN AND Q ITS FIELD OF FRACTIONS

2014 ◽  
Vol 79 (01) ◽  
pp. 296-305 ◽  
Author(s):  
GENA PUNINSKI ◽  
CARLO TOFFALORI
Keyword(s):  
Principal Ideal ◽  
Principal Ideal Domain ◽  
Bezout Domain ◽  
Ideal Domain ◽  
Ziegler Spectrum

Abstract We describe the Ziegler spectrum of a Bézout domain B=D+XQ[X] where D is a principal ideal domain and Q is its field of fractions; in particular we compute the Cantor–Bendixson rank of this space. Using this, we prove the decidability of the theory of B-modules when D is “sufficiently” recursive.

Invariant Factors Under Rank One Perturbations

1980 ◽  
Vol 32 (1) ◽  
pp. 240-245 ◽  
Author(s):  
Robert C. Thompson
Keyword(s):  
Commutative Ring ◽  
Principal Ideal ◽  
Matrix Multiplication ◽  
Diagonal Form ◽  
Principal Ideal Domain ◽  
Ideal Domain ◽  
Zero Divisors ◽  
Rank One ◽  
Invariant Factors

Let R be a principal ideal domain, i.e., a commutative ring without zero divisors in which every ideal is principal. The invariant factors of a matrix A with entries in R are the diagonal elements when A is converted to a diagonal form D = UAV, where U, V have entries in R and are unimodular (invertible over R), and the diagonal entries d1 …, dn of D form a divisibility chain: d1|d2| … |dn. Very little has been proved about how invariant factors may change when matrices are added. This is in contrast to the corresponding question for matrix multiplication, where much information is now available [6].

scholarly journals Strictly real Banach algebras

1993 ◽  
Vol 47 (3) ◽  
pp. 505-519 ◽  
Author(s):  
John Boris Miller
Keyword(s):  
Banach Algebra ◽  
Uniqueness Theorem ◽  
Banach Algebras ◽  
Complex Algebra ◽  
Conjugation Operator ◽  
Complex Banach Algebra

A complex Banach algebra is a complexification of a real Banach algebra if and only if it carries a conjugation operator. We prove a uniqueness theorem concerning strictly real selfconjugate subalgebras of a given complex algebra. An example is given of a complex Banach algebra carrying two distinct but commuting conjugations, whose selfconjugate subalgebras are both strictly real. The class of strictly real Banach algebras is shown to be a variety, and the manner of their generation by suitable elements is proved. A corollary describes some strictly real subalgebras in Hermitian Banach star algebras, including C* algebras.

scholarly journals Finitely generated cyclic extensions of free groups are residually finite

1971 ◽  
Vol 5 (1) ◽  
pp. 87-94 ◽  
Author(s):  
Gilbert Baumslag
Keyword(s):  
Free Group ◽  
Principal Ideal ◽  
Free Groups ◽  
Cyclic Extension ◽  
Principal Ideal Domain ◽  
Finitely Generated ◽  
Finitely Generated Module ◽  
Residually Finite ◽  
Ideal Domain ◽  
Direct Sum

We establish the result that a finitely generated cyclic extension of a free group is residually finite. This is done, in part, by making use of the fact that a finitely generated module over a principal ideal domain is a direct sum of cyclic modules.

scholarly journals Decomposition of representation algebras

1969 ◽  
Vol 10 (3-4) ◽  
pp. 395-402 ◽  
Author(s):  
W. D. Wallis
Keyword(s):  
Finite Group ◽  
Principal Ideal ◽  
Isomorphism Class ◽  
Principal Ideal Domain ◽  
Ideal Domain ◽  
A Finite Group

Throughout this paper g is a finite group and f is a complete local principal ideal domain of characteristic p where p divides |g|. The notations of [5] are adopted; moreover we shall denote the isomorphism-class of an f g-representation module ℳ by M, the class of ℳx by Mx and the class of ℳR by MR for suitable groups K and R.

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