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.
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$\begingroup$ So since the variables are the only irreducible monomials in a polynomial ring, these are the only possible generators for prime monomial ideals? $\endgroup$– SophieDec 7, 2014 at 14:49
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1 Answer
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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 $$I = (x_{i_1}, \ldots \ ,x_{i_k})$$
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2$\begingroup$ 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 $k$. $\endgroup$ Dec 11, 2014 at 16:37