A constructive proof of Orzech’s theorem

Corollary 0.2 is a well-known fact (e.g., it appears in [12, Lemma A.3] and in [3]), but most of its proofs in literature do not generalize to Theorem 0.1. Orzech’s original proof of Theorem 0.1 (with the corrections provided in [2], as the original version was shaky) proceeds by reducing the theorem to the case when A is Noetherian, and then using this Noetherianness in an elegant and yet mysterious way. The proof is not constructive and (to my knowledge) cannot easily be made constructive. In this note, I will present a constructive way to prove Theorem 0.1. Let us first make some preparations. We let N = {0, 1, 2, . . .}. We fix a commutative ring A with unity. For every n ∈N, let In denote the identity n× n-matrix in An×n. We reserve a fresh symbol X as an indeterminate for polynomials. We embed A into the polynomial ring A [X] canonically, and we use this to embed the matrix ring An×n into (A [X])n×n canonically for every n ∈ N. For every n ∈ N and any square matrix M ∈ An×n, we define the characteristic polynomial χM of M as the polynomial det (X · In −M). (This is one of the two common ways to define a characteristic polynomial of a matrix M. The other way is to define it as det (M− X · In). These two definitions result in two polynomials which differ only by multiplication by (−1).) The famous Cayley-Hamilton theorem states the following: