(I think we should mention a commutative ring with unity first).

We need to show that,

is not equal to some

Let us assume that it is,

thus,

since 2 is prime and

are polynomials we conclude they must be constant and that

,

.

Thus, if it is an ideal it must be either,

But,

And

contains all polynomials with the constant term being even, thus,

Thus, if it is an ideal it must be,

But,

contains all the polynomials with even coefficients. While

does contains some that are not, for example,

. Thus,

An impossibility.

(What we actually shown is that,

)

Now, what ails me is that you say,

"conclude Z[x] is not a principal ideal ring"

That is not true, any commutative ring with unity always is a principal ideal ring. What you probabaly wanted to say is that, "conclude that (2,x) is not a principal ideal ring in Z[x]".

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That is impossible, the kernel of any ring homomorphism is an ideal!

is not an ideal in

.