# Dedekind-finite ring(right invertibles are left invertibles) different from ring that left invertibles are right invertibles?

Let $R$ be a unital ring s.t. $R$ is dedekind finite.(i.e. (1) If $a$ is right invertible, then $a$ is left invertible.)

Consider (2) If $a$ is left invertible, then $a$ is right invertible.

I do not see any good reason that $(1)\implies (2)$ or $(2)\implies (1)$. Of course, if an element has both left and right inverse, then the inverse coincides.

$\textbf{Q:}$ Are $(1)$ and $(2)$ equivalent? The book defines $R$ dedekind finite if $a$ right invertible$\implies a$ left invertible. What if I switch left to right by saying $a$ is left invertble$\implies a$ right invertible. Why do I prefer the definition of dedekind finite to (2)?

Ref: A first course in non-commutative rings by T.Y.Lam Chpt 1.

Suppose $ba=1$ for some $b$. Then $b$ is right invertible, and by (1), also left invertible. But then $a$ is also the left inverse of $b$. (By the usual argument for the uniqueness of the inverse: if $cb=1$ then $c=c(ba)=(cb)a=a$.) Hence $ab=1$ and $a$ is also right invertible.