Chapter 5.1, Problem 9E

Exercises

Prove Theorem 5.3:A subset $S$

of the ring $R$

is a subring of $R$

if and only if these conditions are satisfied:

$S$ is nonempty.

$x\in S$

and $y\in S$

imply that $x+y$

and $xy$ are in $S$.

$x\in S$

implies $-x\in S$.