RO-NMO 2014 Problem 121

For a ring $A,$ and an element $a$ of it, define $s_a,d_a:A\longrightarrow A, s_a(x)=ax,d_a=xa.$ a) Prove that if $A$ is finite, then $s_a$ is injective if and only if $d_a$ is injective.b) Give example of a ring which has an element $b$ for which $s_b$ is injective and $d_b$ is not, or, conversely, $s_b$ is not injective, but $d_b$ is.