# Answer to Question #17116 in Abstract Algebra for sanches

Question #17116

Let R be any semisimple ring. If a ∈ R is such that I = aR is an ideal in R, then I = Ra.

Expert's answer

By the Wedderburn-Artin Theorem, we are reduced to the case when

Since

*R*= M*n*(*D*) where*D*is a division ring.Since

*R*= M*(*_{n}*D*) is now a simple ring, we have either*I*= 0 or*I*=*R*. If*I*= 0, then*a*= 0 and*Ra*=*I*holds. If*I*=*R*, then*ab*= 1 for some*b**∈**R*. By Dedekind-finiteness of*R*,*ba*= 1, so*Ra*=*I*also holds.
## Comments

## Leave a comment