R-endomorphisms of R[[X]] are essentially continuous

Paul M. Eakin, Jr. and Avinash Madhav Sathaye

Vol. 66 (1976), No. 1, 83–87

DOI: 10.2140/pjm.1976.66.83

Abstract

Let R be a commutative ring with identity, A = R[[X]] and B = R[[Y ]] with X and Y finite sets of indeterminates. Consider A and B as topological rings with the respective X and Y -adic topologies. If σ : A → B is any R-homomorphism then there are R-automorphisms s and t of A and B respectively, so that t ∘ σ ∘ s : A → B is continuous. As a corollary we see that an R-endomorphism of A is surjective only if it is an automorphism.

Mathematical Subject Classification 2000

Primary: 13J05

Milestones

Received: 28 July 1975

Published: 1 September 1976