Polynomial Ring, Show I is prime but not maximal

In summary, in this conversation, the topic of discussion was the principal ideal of a polynomial ring R = Z[x], where Z is the set of integers, generated by x. It was proven that this ideal, denoted by I, is a prime ideal of R but not a maximal ideal. It was suggested to show that R/I is an integral domain and not a field, or to find a ring that is isomorphic to R/I and prove it through an isomorphism. Additionally, it was recommended to simplify things by considering specific polynomials and their representations in R/I, or to try doing arithmetic in R/I.
  • #1
Rederick
12
0

Homework Statement



Let R = Z[x] be a polynomial ring where Z is the integers. Let I = (x) be a principal ideal of R generated by x. Prove I is a prime ideal of R but not a maximal ideal of R.

Homework Equations


The Attempt at a Solution



I want to show that R/I is an integral domain which implies I is a prime ideal and that R/I is NOT a field which implies I is not a maximal ideal.
I'm not sure how to represent R/I to show those two things. I know R/I = f(x) +I but I don't know where to go from there.
 
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  • #2
Rederick said:
I'm not sure how to represent R/I to show those two things. I know R/I = f(x) +I but I don't know where to go from there.
For any particular polynomial f, the element represented by f(x) + I also has many other representations as g(x) + I for other polynomials g. It is often helpful to simplify things; maybe it will suggest something.

An alternative would be to guess a ring that is isomorphic to the quotient R/I, and try to write down the isomorphism, then try to prove both directions are well-defined.

If all else fails, you could try doing arithmetic in R/I to gain familiarity with it, or maybe try to prove directly from the definitions of everything involved that R/I is an integral domain that is not a field.
 

FAQ: Polynomial Ring, Show I is prime but not maximal

What is a polynomial ring?

A polynomial ring is a mathematical structure that consists of polynomials with coefficients from a certain field or ring. It is denoted as R[x], where R is the ring and x is an indeterminate or variable.

What does it mean for a polynomial ring to be prime?

A polynomial ring R[x] is prime if and only if R is a commutative ring with unity and the ideal generated by x is a prime ideal. This means that x is a prime element in R[x], and any polynomial in R[x] that is not divisible by x is also a prime element.

What is the difference between a prime and maximal ideal in a polynomial ring?

A prime ideal is an ideal in a polynomial ring that is not the whole ring and has the property that whenever the product of two polynomials is in the ideal, at least one of the polynomials is in the ideal. A maximal ideal, on the other hand, is an ideal that is not contained in any other proper ideal. In other words, a maximal ideal cannot be properly contained in any other ideal.

How can you show that a polynomial ring is prime but not maximal?

To show that a polynomial ring is prime but not maximal, you can use the definition of prime and maximal ideals and check if the ideal generated by x is prime and if there is any ideal containing it that is not the whole ring. If both conditions are satisfied, then the polynomial ring is prime but not maximal.

What are some examples of polynomial rings that are prime but not maximal?

Some examples of polynomial rings that are prime but not maximal are the polynomial ring Z[x] over the integers, the polynomial ring R[x] over a commutative ring R, and the polynomial ring F[x] over a field F. In all of these examples, the ideal generated by x is prime but not maximal.

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