Answer to Question #17366 in Algebra for Melvin Henriksen

Question

Show that any unit-regular ring R is Dedekind-finite.

Answer 1

Suppose ab = 1 ∈ R, where R is unit-regular. Write a =aua, where u ∈ U(R). Then 1 = ab = auab= au, so a = u^−1 ∈ U(R).