Let $E$ be a ring.
And let $N$ be the set of non-units. Assume that it is a additive subgroup of $E$(so $E$ is local).
I need to prove $N$ is two-sided ideal and the commutative case is trivial.
For the non-commutative case I have that $N$ is mult. closed and multiplying a (two-sided)unit with a non-unit yields a non-unit. But it gets difficult for me when you look at elements with a one-sided inverse.
Assume $b$ only has a right inverse and let $u\in N$. Then, why, for all $a$ with only a left inverse:
Since this is required if $N$ is a ideal. Any hints? Where can I use the additive subgroup assumption?