This is perhaps a very trivial question, but I just want to be sure.
Suppose that, by assumption, some element of a ring is invertible. Rings need not be closed under multiplicative inverses, surely, but is it reasonable or necessary to conclude that this inverse is also an element of the ring?
I've been trying to piece together an example wherein this is guaranteed by closure under multiplication or the fact that, if $y$ is the inverse of some $x \in R$, then clearly $x$ is also the inverse of $y$, but this doesn't seem to guarantee anything, unless I'm missing something obvious.
Thanks in advance.