# Prime ideal

• Apr 24th 2010, 08:44 AM
empressA88
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.
• Apr 24th 2010, 08:53 AM
tonio
Quote:

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
• Apr 24th 2010, 08:58 AM
empressA88
$\displaystyle x-1,\,x^2+1\notin <x^3-x^2+x-1>$

Tonio[/QUOTE]

I understand everything else besides this part.
• Apr 24th 2010, 09:04 AM
tonio
Quote:

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
• Apr 24th 2010, 09:25 AM
empressA88
ok..let me see if am on the same page..the principal ideals are x-1, x^2 + 1?
• Apr 24th 2010, 09:28 AM
empressA88
ok..i get where you are coming from. Sorry..thank you