Ideal generated by two prime elements in a unique factorization domain may be proper and not prime
From Commalg
Contents
Statement
It is possible to have a unique factorization domain and primes
such that
is a proper ideal of
that is not prime.
Related facts
- Unique factorization is not prime-quotient-closed: If
is a unique factorization domain, and
is a prime ideal, the quotient
need not be a unique factorization domain. Examples of this where the prime ideal
is principal also give examples of ideals generated by two elements that are proper and not prime.
Opposite facts
- Unique factorization implies every prime ideal is generated by prime elements
- Unique factorization implies every nonzero prime ideal contains a prime element
- Unique factorization and Noetherian implies every prime ideal is generated by finitely many prime elements
Proof
Example of a bivariate polynomial ring
Let be any field of characteristic not equal to
. Let <mah>R = k[x,y]</math>. The polynomials
and
are irreducible in
, and hence prime. However, the ideal
is not prime in
. To see this, note that
, which is not an integral domain, because
is not irreducible.
Example of a polynomial ring over the integers
Let be the ring of rational integers and
. The elements
are irreducible in
. However, the ideal
is not a prime ideal in
. To see this, note that
being prime is equivalent to the polynomial
being irreducible in
, which is not true since the polynomial reduces as
modulo
.