A short proof of Grinshpon's theorem

TL;DR

This paper provides a concise proof of Grinshpon's theorem, which states that if a matrix over a commutative subring is invertible in a larger ring, then its determinant is invertible in that larger ring.

Contribution

The paper introduces a significantly shorter proof of Grinshpon's theorem, simplifying the understanding of invertibility conditions for matrices over rings.

Findings

Short proof of Grinshpon's theorem presented

Clarifies the relationship between invertibility of matrices and determinants in ring extensions

Simplifies existing proofs for better comprehension

Abstract

Grinshpon has proved that if $S$ is a commutative subring of a ring $R$ and $A\in M_n(S)$ is invertible in $M_n(R)$, then $det(A)$ is invertible in $R$. We give a very short proof of the result.