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

Question

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.

Answer 1

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 $$I = (x_{i_1}, \ldots \ ,x_{i_k})$$