Rank 1 preserving maps on linear transformations over noncommutative local rings
type: Item
instance of: scholarly article
title: Rank 1 preserving maps on linear transformations over noncommutative local rings
zbMATH Open document ID: 0661.15012
DOI: 10.1016/0021-8693(88)90160-3
author: Warren J. Wong
published in: Journal of Algebra
publication date: 1988-01-01T00:00:00Z
review text: 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.
Mathematics Subject Classification ID: 15A30
Mathematics Subject Classification ID: 16L30
zbMATH DE Number: 4079631
zbMATH Keywords: matrix algebra
zbMATH Keywords: commutative ring
zbMATH Keywords: local ring
zbMATH Keywords: K-module
zbMATH Keywords: K-linear maps
MaRDI profile type: Publication
full work available at URL: 0021-8693(88)90160-3
OpenAlex ID: W2039418106
cites work: Separable algebras over commutative rings
cites work: Localization in non-commutative Noetherian rings
cites work: Q3882586
cites work: Projective modules
cites work: Q5343374
cites work: 𝑅-linear endomorphisms of (𝑅)_{𝑛} preserving invariants
cites work: Introduction to Algebraic K-Theory. (AM-72)
cites work: Automorphisms of \(\det (X_{ij}):\) The group scheme approach
description: scientific article; zbMATH DE number 4079631