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RING ENDOMORPHISMS WITH LARGE IMAGES

Published online by Cambridge University Press:  25 February 2013


ANDRÉ LEROY
Affiliation:
Université d'Artois, Faculté Jean Perrin Rue Jean Souvraz 62 307 Lens, France e-mail: leroy@euler.univ-artois.fr
JERZY MATCZUK
Affiliation:
Institute of Mathematics, Warsaw University Banacha 2, 02-097 Warsaw, Poland e-mail: jmatczuk@mimuw.edu.pl
Corresponding

Abstract

The notion of ring endomorphisms having large images is introduced. Among others, injectivity and surjectivity of such endomorphisms are studied. It is proved, in particular, that an endomorphism σ of a prime one-sided noetherian ring R is injective whenever the image σ(R) contains an essential left ideal L of R. If, in addition, σ(L)=L, then σ is an automorphism of R. Examples showing that the assumptions imposed on R cannot be weakened to R being a prime left Goldie ring are provided. Two open questions are formulated.


Type
Research Article
Copyright
Copyright © Glasgow Mathematical Journal Trust 2013

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