Rank 1 preserving maps on linear transformations over noncommutative local rings
Many authors have studied the problem of determining the linear maps on the \(n\times n\) matrix algebra \(M_ n(K)\) over a commutative ring K which preserve various properties. Frobenius characterized the maps leaving the determinant function unchanged, in the case \(K={\mathbb{C}}\). \textit{B. R. McDonald} [Mem. Am. Math. Soc. 287 (1983; Zbl 0529.15011)] and \textit{W. C. Waterhouse} [Automorphisms of det \((x_{ij}):\) The group scheme approach, Adv. Math. 65, 171-203 (1987)] solved the problem of finding maps leaving the set of rank one matrices invariant in the case of arbitrary commutative ring K. The present author considers a more general problem. Let R be a local ring, not necessarily commutative, and R be an algebra over a commutative ring K, such that R is a finitely generated K- module. The K-linear maps on \(M_ n(R)\) which leave the set of rank one matrices invariant are determined.
scientific article; zbMATH DE number 4079631