International Journal of Mathematics and Mathematical Sciences
Volume 7 (1984), Issue 2, Pages 297-301

Rings all of whose additive group endomorphisms are left multiplications

Michael I. Rosen1 and Oved shisha2

1Department of Mathematics, Brown University, Providence 02912, RI, USA
2Department of Mathematics, University of Rhode Island, Kingston 02881, RI, USA

Received 6 October 1983

Copyright © 1984 Michael I. Rosen and Oved shisha. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


Motivated by Cauchy's functional equation f(x+y)=f(x)+f(y), we study in §1 special rings, namely, rings for which every endomorphism f of their additive group is of the form f(x)ax. In §2 we generalize to R algebras (R a fixed commutative ring) and give a classification theorem when R is a complete discrete valuation ring. This result has an interesting consequence, Proposition 12, for the theory of special rings.