Prove that every ideal of Zn is principal. Is Zn principal? Solution If J = {0} then J = I(0). Otherwise there exists a := spe(J). Let X := J I(a). If X is non-empty then there exists x := spe(X). Of course a = x and 0 = x – a ? X. Since x was the smallest positive element of X, and a > 0, this means that x – a = 0, i.e. x = a. But x ? I(a). This contradiction shows that X is empty. Thus J = I(a). END of proof Ideals of the form I(a) are called principal. Thus every ideal in Z is principal. On the top of it Z is an integral domain. For these reasons Z is called a principal ideal domain or a PID for short. Examples of ideals: * I(0) = {0} – the zero ideal * I(1) = Z – the whole ring.