Matrices that are integral over the base ring.

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]