# When is the quotient ring of a multivariable polynomial ring over a field by a monomial ideal an integral domain?

When is the quotient ring of a multivariable polynomial ring over a field by a monomial ideal an integral domain?

I am actually trying to show that a monomial ideal is prime by showing the corresponding quotient ring is an integral domain.

• So since the variables are the only irreducible monomials in a polynomial ring, these are the only possible generators for prime monomial ideals? – Sophie Dec 7 '14 at 14:49

## 1 Answer

A monomial ideal in $k[x_1, \ldots x_n]$ with $k$ a field is prime if and only if is of the following type $$I = (x_{i_1}, \ldots \ ,x_{i_k})$$

• Can you suggest any references? – Sophie Dec 7 '14 at 14:51
• In fact, the proof is somewhat trivial. Suppose your ideal $I$ is prime and let $m=x_{i_1}\cdots x_{i_k}$ a monomial in it; we want to show that one of the variables appearing in $m$ belongs to $I$. Do this by induction on $k$. – Mariano Suárez-Álvarez Dec 11 '14 at 16:37