(a) Suppose that
is an integral domain of positive characteristic. Suppose that
is an ideal. Prove that
has the same characteristic as
.
Hints - Try to show the following:
1) The characteristic of any integral domain is either zero or a prime
2) If for an abelian group G we define , then
the exponent of any subgroup of G is a divisor of the group's exponent or zero if the
latter is zero.
(b) Give an example which demonstrates that
need not be an integral domain.