# Thread: Irreducible but not prime

1. ## Irreducible but not prime

I need an example of a ring that is irreducible but not prime.

there is this example of such a ring in the book saying in Z[-(root(3)] where
4= 2*2
but also 4= (1+root(-3))*(1-root(-3))
okay I get it that and I know 1+root(-3) doesn't divide 2 hence is is not prime. but also if it was reducible then we should have units
since the definition of irreducible is that a=b*c then b or c is unit. what is the unit of this ring !!!!?

(ps: if one day I understand everything in abstract algebra, surely I will rewrite the book)

2. Originally Posted by hamidr
I need an example of a ring that is irreducible but not prime.

there is this example of such a ring in the book saying in Z[-(root(3)] where
4= 2*2
but also 4= (1+root(-3))*(1-root(-3))
okay I get it that and I know 1+root(-3) doesn't divide 2 hence is is not prime. but also if it was reducible then we should have units
since the definition of irreducible is that a=b*c then b or c is unit. what is the unit of this ring !!!!?
You mean an irreducible element but not a prime element in the domain?

You can find such elements in a non-UFD. In UFD, every irreducible element is a prime element though.

Now, take some non-UFD examples.

Consider $\displaystyle D = F[x^2, xy, y^2]$, where F is a field. Then, $\displaystyle (x^2)(y^2) = (xy)(xy)$ (Note: the parenthesis here is not denoted for an ideal ). We see that $\displaystyle (xy) | (x^2)(y^2)$, but it does not imply either $\displaystyle (xy) |(x^2)$ or $\displaystyle (xy) | (y^2)$. Thus xy is not a prime element. However, xy is an irreducible element in D. For example, if $\displaystyle F = \mathbb{Q}, xy = (3xy) \times (1/3)$, then you see that 1/3 is a unit here and xy is not factorized $\displaystyle (x)(y)$ in this domain.

As per your example, $\displaystyle 1+\sqrt{-3}$ is not a prime element, but it is an irreducible element. You cannot factorize $\displaystyle 1+\sqrt{-3}$ = ab unless a or b is a unit in $\displaystyle \mathbb{Z}[\sqrt{-3}]$.

(ps: if one day I understand everything in abstract algebra, surely I will rewrite the book)

3. Your example (4=2*2=(1+sqrt(-3))(1-sqrt(-3))) works, but the reason is not correct. You should show that neither of them differ by a unit, and everything in the factorization (2,1+sqrt(-3),1-sqrt(-3)) is irreducible.

The unit in Z[sqrt(-3)] is just 1 and -1, since you can define a positive-integer-valued multiplicative "norm" on non-zero elements, and it's easy to show that a unit must have a norm 1, while the only elements in Z[sqrt(-3)]* having norm 1 are 1 and -1.

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# irreducible element that is not prime

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