A problem about invertibility in an unitarian ring Let $(R,+,*)$ be an unitarian ring. (a) If $A$ is an unitarian ring. $a,b$ are from $R$ and $a*b$ is invertible, it results that $a$ is invertible and $b$ is invertible? (b) If $a$ is from $R$ and $a^n=a * a*...* a$ ($n$ or) is invertible,it results that $a$ is invertible? (c) If $a$ is invertible to left and it isn't zero divisor to right,prove that $a$ is invertible. (a) Here I tried to find a counterexample to that affirmation, but I couldn't find it. (b) I wrote the definition of invertibility for a^n,but from that I couldn't find anything. (c) I wrote the definition of invertibility and for non-zero divisibility for $a$, but I couldn't deduce anything from it. Please, could you help me?

Answer on (a):Not in general. See the answers to this question.

Answer on (b):

If $ba^n=1=a^nb$ then $(ba^{n-1})a=1=a(a^{n-1}b)$ showing that $a$ has a left- _and_ a rightinverse. Consequently $a$ is invertible.