Let $R$ be a unital ring and $S\le R$ be a unital subring with $1_S = 1_R$.
Then it can be shown that $S^\times\subset R^\times\cap S$, where $R^\times$ and $S^\times$ are the sets of units of respective rings. (Statement *)
There can be found a ring $R$ and its subring $S$ such that $R^\times\cap S\ne S^\times$. That is, $\exists a\in R^\times\cap S$ such that $a\not\in S^\times$. (Statement **)
I have a couple of questions, as I'm feeling quite confused:
(1) Is it possible that $S\le R$ and yet $1_S\ne 1_R$?
(2) I can't think of an example of a ring and a subring that satisfy the Statement **. Can someone please give an example?