Show that ⟨2,x⟩\langle 2,x \rangle is not a principal ideal in Z[x]\mathbb Z [x]

Hi
I don’t know how to show that 2,x is not principal and the definition of a principal ideal is unclear to me. I need help on this, please.

The ring that I am talking about is Z[x] so 2,x refers to 2g(x)+xf(x) where g(x), f(x) belongs to Z[x].

Answer

I think it’s relatively easy to see that I=2,x={anxn++a1x+a0;a0 is even}.

Now, suppose that I=f(x) for some f(x)I.

If f(x) is a constant polynomial, then f(x) contains only polynomials with even coefficients, and we do not get x.

If f(x) is of degree at least 1, then non-zero polynomials in f(x) have degree at least 1, and we do not get 2.

So I is not of the form f(x).

Attribution
Source : Link , Question Author : Person , Answer Author : Martin Sleziak

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