Suppose $R$ is a ring with unity $1$ and for some $a\in R$ there exists more than one left inverse of $a$ in $R$. Show that $R$ has infinitely many left inverses of $a$.
I am trying to define a map $f:X\to X$ which is one-to-one but not onto or vice versa where $X=\{x\in R\mid xa=1\}$. That will directly imply that cardinality of $X$ can't be finite. But I don't have exact idea of proceeding further.
Edit:Given Ring is non commutative.