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Matrices that are integral over the base ring.
- Source :
- Journal of Algebra & Its Applications; Apr2022, Vol. 21 Issue 4, p1-10, 10p
- Publication Year :
- 2022
-
Abstract
- We study matrices over arbitrary rings that are integral over the base ring. For any ring R , let f , g , h ∈ R [ x 1 , x 2 , ... , x n ] with h = f g. We prove that for any pairwise commuting elements a 1 , a 2 , ... , a n ∈ R if g (a 1 , a 2 , ... , a n) = 0 , then h (a 1 , a 2 , ... , a n) = 0. As a corollary, it follows that for f ∈ n (S) [ x 1 , x 2 , ... , x n ] , S commutative ring, if A 1 , A 2 , ... , A n ∈ n (S) are pairwise commuting matrices such that f (A 1 , A 2 , ... , A n) = 0 , then g (A 1 , A 2 , ... , A n) = 0 where g = det (f) I. This result, which is a generalization of the Cayley–Hamilton Theorem, was proved by Phillips in 1919. For a positive integer n > 1 , we prove that if every matrix in n (R) satisfies a monic polynomial of degree n over R , then R is commutative. On the other hand, every diagonal matrix in n (R) , n > 1 , satisfies a monic polynomial of degree n over R precisely when R is a left duo ring. We prove that if every diagonal matrix in n (R) , n > 1 , is R -integral, then R is Dedekind finite. [ABSTRACT FROM AUTHOR]
- Subjects :
- MATRICES (Mathematics)
COMMUTATIVE rings
FINITE rings
INTEGRALS
POLYNOMIALS
Subjects
Details
- Language :
- English
- ISSN :
- 02194988
- Volume :
- 21
- Issue :
- 4
- Database :
- Complementary Index
- Journal :
- Journal of Algebra & Its Applications
- Publication Type :
- Academic Journal
- Accession number :
- 155815944
- Full Text :
- https://doi.org/10.1142/S0219498822500803