Ring Endomorphisms with Large Images

André Leroy (1) , Jerzy Matczuk (2)

1 LML - Laboratoire de Mathématiques de Lens
2 Institute of Mathematics [Warsaw]

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 S of a prime one-sided noetherian ring R is injective whenever the image S (R) contains an essential left ideal L of R. If additionally S(L) = L, then S is an automorphism of R. Examples showing that the assumptions imposed on R can not be weakened to R being a prime left Goldie ring are provided. Two open questions are formulated.

Mots clés: Endomorphisms of semi-prime rings; Hopfian co-Hopfian rings

Domaines: Anneaux et algèbres [math.RA]

André Leroy, Jerzy Matczuk. Ring Endomorphisms with Large Images. 2012. ⟨hal-00718394⟩