Why is <2>[x] not maximal in Z[x] if <2> is maximal in Z?

[wowolala]

here is the quesiton:

let I=<2>. prove that I[x] is not a maximal ideal of Z[x] even though I is a maximal ideal of Z

can someone help me?

 

[morphism]

What is I[x] supposed to be?

 

[HallsofIvy]

What is I[x] supposed to be?

Morphism, are you asking this to make sure that wowolala understands it or can anyoe answer?

 

[morphism]

Anyone can answer - but who's to tell if what they say is what wowolala had intended. It's just that I'm not familiar with the notation I[x] when I is an ideal, probably because for me rings have always been unital so that ideals aren't necessarily (sub)rings. But I guess I[x] is the ideal generated by I and {x}?

 

[HallsofIvy]

My "guess" would be that, since I is the set of all even integers, I[x] is the set of all polynomials, in the variable x, having even integers for coefficients. That is an ideal of Z[x] but not maximal: there is no single polynomial, having even coefficients, that generates the ideal.

 

[morphism]

there is no single polynomial, having even coefficients, that generates the ideal.

That's true, but it proves that I[x] isn't principal and not that it isn't maximal.

I think the fastest way to prove non-maximality is to observe that Z[x]/I[x] isn't a field.

 

[HallsofIvy]

Oh, dear, I misread the question!