scholarly journals The Structure of Symmetric Solutions of the Matrix Equation AX=B over a Principal Ideal Domain

2017 ◽  
Vol 2017 ◽  
pp. 1-7
Author(s):  
V. M. Prokip
Keyword(s):  
Matrix Equation ◽  
Principal Ideal ◽  
Symmetric Solution ◽  
Principal Ideal Domain ◽  
Ideal Domain ◽  
The Matrix

We investigate the structure of symmetric solutions of the matrix equation AX=B, where A and B are m-by-n matrices over a principal ideal domain R and X is unknown n-by-n matrix over R. We prove that matrix equation AX=B over R has a symmetric solution if and only if equation AX=B has a solution over R and the matrix ABT is symmetric. If symmetric solution exists we propose the method for its construction.

scholarly journals On solvability of the matrix equation AXB = C over a principal ideal domain

2020 ◽  
pp. 47-50
Author(s):  
Volodymyr Prokip
Keyword(s):  
Matrix Equation ◽  
Principal Ideal ◽  
Normal Forms ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Principal Ideal Domain ◽  
Ideal Domain ◽  
The Matrix ◽  
Smith Normal Forms ◽  
Necessary And Sufficient

In this paper we present conditions of solvability of the matrix equation AXB = B over a principal ideal domain. The necessary and sufficient conditions of solvability of equation AXB = B in term of the Smith normal forms and in term of the Hermi-te normal forms of matrices constructed in a certain way by using the coefficients of this equation are proposed. If a solution of this equation exists we propose the method for its construction.

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.

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 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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