$R$ is integral domain

Question

Let$ S$ be a subring of$ R$. If $R$ is an integral domain, then $1_R$ = $1_S$?

I know that if $S$ is subring of $R$ then $S$ is integral domain. How to prove this problem ?

Thanks in advance.

Sorry, found an even better duplicate here. You could have found them by using the search feature with terms like "domain subring identity". — rschwieb, Mar 18 '18 at 11:12

score 3 · Answer 1 · answered Mar 18 '18 at 06:02

3

Observe that

$1_S(1_R - 1_S) = 1_S 1_R - 1_S^2 = 1_S - 1_S^2 = 1_S - 1_S = 0; \tag 1$

then since $R$ is an integral domain, if

$1_S \ne 0, \tag 2$

we must have

$1_R - 1_S = 0, \tag 3$

which implies

$1_R = 1_S. \tag 4$

answered Mar 18 '18 at 06:02

Robert Lewis

1 Answers1