Question

Show that a ring R is not Dedekind-finite iff there exists an idempotent e in R such that eR isomorphic R as right R-modules.

Answer 1

R is not Dedekind-finite iff there exists an R-isomorphism $R_{R} \cong R_{R} \oplus X$ for some R-module $X \asymp 0$. This means that $R = eR \oplus (1 - e)R$ for some idempotent $e \asymp 1$ such that $\left(eR\right)_R \cong R_R$.