# Problem related to unit of a ring

Recently, I have encountered a problem:

Given a ring $R$. Prove that:

1) If $x \in R$ has two distinct right inverses, it has infinitely many right inverses

2) If $x \in R$ has a unique right inverse, it is invertible.

3) If $R$ doesn't have any zero divisors, every left or right unit is a unit.

For the first task, I can only see an example: Given ring $R$ with the multiplication $x . y = x$, then it is easily seen that $1$ has infinitely many right inverses. However, it doesn't provide any insights for me to reach the solution.

For the second task, I guess that if $x$ has the unique right inverse $y$, then $y$ must also be the inverse of $x$. However, I don't have a way to get it.

For the third task, I think that it is not necessary that the left/right inverse of $x$ must be the inverse of $x$. However, for this case, I can't go any further.

Please consider the problem and give me some hints. Any help is appreciated. Thank you for reading.

(2). Suppose that $xa = 1$. Then $xax = x$, whence $x(ax-1) = 0$ and $x(a+(ax-1)) = 1$. Since $x$ has a unique right inverse, $a = a + (ax-1)$ and thus $ax = 1$. Thus $a$ is an inverse of $x$.
(3) Same proof, but use the fact that there is no zero divisor to conclude directly that $ax - 1 = 0$.
• Thank you, sir. However, there is still one thing I'm wondering about the proof for (1). I can't prove the fact that if $x \in R$ has a right inverse and a left inverse, then $x$ has a unique inverse. Please show me. Commented Dec 31, 2017 at 6:28
• Also, I'm still working on how the author found out the expression $a + (1 - ax)x^n$. I can "temporally" explain that because $x(1 - ax) = x - xax = x - x.1 = 0$, so $x(1 - ax)x^n = 0$, thus $xa + x(1 - ax)x^n = 1$. But starting from scratch, is there a way to find this out without relying on experience? Commented Dec 31, 2017 at 6:31
• The proof shows that if $a$ is any left inverse and $b$ is any right inverse, then $a=b$. If $c$ is any other right inverse, then $c=a$ so $c=b$. The proof really shows that if there is a a right inverse and a left inverse, then it is unique. Commented Jan 1, 2018 at 3:56