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Show that $I=(X_{k_1}^{a_1},...,X_{k_s}^{a_s})$ is $(X_{k_1},...,X_{k_s})$-primary, where $I$ is the ideal generated by the monomials $X_{k_1}^{a_1},...,X_{k_s}^{a_s}$ .

I noticed that $$\sqrt{I}=\sqrt{({X_{i_1}}^{a_1},...,{X_{i_k}}^{a_k})}=\sqrt{({X_{i_1}}^{a_1})+...+({X_{i_k}}^{a_k})}=\sqrt{\sqrt{(X_{i_1})^{a_1}}+...+\sqrt{(X_{i_k})^{a_k}}}=\sqrt{(X_{i_1})+...+(X_{i_k})}=\sqrt{(X_{i_1},...,X_{i_k})}=(X_{i_1},...,X_{i_k}).$$ I have to show that $I$ is primary, i.e. if $u\cdot v\in I$ then $u\in I$ or $v\in (X_{k_1},...,X_{k_s})$.

user26857
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Ancastack
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  • You might take some time improving your question. Is this a homework problem? –  Apr 20 '16 at 15:01
  • No! I want to understand primary decompositions of monomial ideals. – Ancastack Apr 20 '16 at 15:03
  • @ChrisRamsey is this question too easy? – Ancastack Apr 20 '16 at 15:22
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    Not sure as it isn't my area, but whenever someone says "show" it often is homework. I would edit the question specifying what the terms are and add something about why you are interested in this. –  Apr 20 '16 at 15:28

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