91Ó°ÊÓ

Every subring of a field is an integral domain.

Explain why a field \(F\) can have no nontrivial ideals (that is, no ideals except \(\\{0\\}\) and \(\mathrm{F}\) ).

Give an example of a subring of \(P_{3}\) which is not an ideal.

The center of a ring \(A\) is the set of all the elements \(a \in A\) such that \(a x=x a\) for every \(x \in A\). Prove that the center of \(A\) is a subring of \(A\).