Algebra.....units prove

Quote:

Originally Posted by suedenation

Suppose that R and S are rings with identity. Let (r, s) in R × S. Show that
(r, s) is unit in R × S <=> r is a unit in R and s is a unit in S.

Also, if (r, s) is a unit in R×S, show that inverse of (r, s) = (inverse of r, inverse of s)
(Hint: It is efficient to prove the last statement and the implication “<=” in the first statement at the
same time.)

um....can anyone help!!! urgent!!! Thanks very much :)

If $S,R$ are rings with unity, then their direct product is also a ring $S\times R$ . Let $(s,r)\in S\times R$ . Then, $s\in S$ and $r\in R$ . If $(s,r)$ is a unit there exists, $(s^{-1},r^{-1})$ but $s^{-1}\in S$ and $r^{-1}\in R$ . Now assume the converse, if $s\in S$ and $r\in R$ are units, then there exists $s^{-1}\in S$ and $r^{-1}\in R$ , thus, $(s^{-1},r^{-1})\in R\times S$ is a unit.
Q.E.D.