# Thread: Prime ideal

1. ## Prime ideal

I need help with starting this problem. It says Show that the principal ideal in R[x] generated by x^3 - x^2 + x - 1 is not prime in R[x].

I know that a prime ideal I in a ring R is an ideal such that a,b are elements of R and ab is an element of I implies that either a or b is an element of I. But how do I apply this to this problem.

2. Originally Posted by empressA88
I need help with starting this problem. It says Show that the principal ideal in R[x] generated by x^3 - x^2 + x - 1 is not prime in R[x].

I know that a prime ideal I in a ring R is an ideal such that a,b are elements of R and ab is an element of I implies that either a or b is an element of I. But how do I apply this to this problem.

Note that $\displaystyle x^3-x^2+x-1=(x-1)(x^2+1)\Longrightarrow (x-1)(x^2+1)\in <x^3-x^2+x-1>$ but $\displaystyle x-1,\,x^2+1\notin <x^3-x^2+x-1>$

Tonio

3. $\displaystyle x-1,\,x^2+1\notin <x^3-x^2+x-1>$

Tonio[/QUOTE]

I understand everything else besides this part.

4. Originally Posted by empressA88
$\displaystyle x-1,\,x^2+1\notin <x^3-x^2+x-1>$

Tonio
I understand everything else besides this part.[/quote]

The polynomials $\displaystyle x-1\,\,\,and\,\,\,x^2+1$ do not belong to the ideal generated by $\displaystyle x^3-x^2+x-1$ ...this is very easy to check if you first characterize the pol's that do belong to the ideal.

Tonio

5. ok..let me see if am on the same page..the principal ideals are x-1, x^2 + 1?

6. ok..i get where you are coming from. Sorry..thank you