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Hidden Matrices
Author(s) -
Levy L. S.,
Robson J. C.,
Stafford J. T.
Publication year - 1994
Publication title -
proceedings of the london mathematical society
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.899
H-Index - 65
eISSN - 1460-244X
pISSN - 0024-6115
DOI - 10.1112/plms/s3-69.2.277
Subject(s) - subring , mathematics , ring (chemistry) , combinatorics , reduced ring , principal ideal ring , matrix ring , quaternion , integer (computer science) , simple ring , noncommutative ring , scalar (mathematics) , discrete mathematics , commutative ring , pure mathematics , invertible matrix , commutative property , geometry , chemistry , organic chemistry , computer science , programming language
We investigate subrings of an n × n matrix ring which, despite appearing otherwise, are themselves full rings of n × n matrices; that is, are hidden matrices . In general, this problem is subtle, but we give fairly complete results in a number of situations. For example, we prove: Theorem A. Let K be an ideal of a ring R and suppose that T = (R ij ) is a tiled subring of M n (R) containing M n (K). Suppose that R ii = R jj for all i and j and that R ii /K ≅ M n (D), for some ring D. Then T ≅ M n (S), for a ring S that we describe explicitly. The subtleties are illustrated by the following theorem: Theorem B. Let H denote the ring of integer quaternions and let p be an odd prime number. Set R = H + M 2 (pH), where H is identified with the ring of scalar matrices inside M 2 (H). Then R≅M 2 (S), for some ring S, if and only if p ≡1 (mod 4).