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*