Subring Proof Let R be a ring and S a subset of R Show that

Subring Proof:

Let R be a ring and S a subset of R. Show that 5 is a subring of R if and only if each of the following conditions is satisfied. rs e S for all r, s e S. r - s e S for all r, s e S.

Solution

Suppose S is a sub ring then 0 is in S addition and closure must satify if any a €S then -a€S.so a ,b€S then -b€S a+(-b) €S (closure property).

a.b€S(multiplication closure property)

Now supose set S satisfy above 3 property

We are going to prove S is subring of R

any a€S then a-a=0€S (by c)

0-a=-a€S addition invese law hold

a,b€S then a.b€S ( multiplication closure law hold)

S is subring both side proved