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Definition: An ideal is said to be monomial ideal if it is generated by monomials. For example, in $k[x, y]$ $(xy, y^2)$ is a monomial ideal.

Result: Let $I$ be a monomial ideal with a generator $ab$ (where $a$ and $b$ are coprime), say $I= (ab) +J$, then $ I= ((a) +J) \cap ((b) +J))$.

Hence $ (xy, y^2) =(x, y^2) \cap (y, y^2) $ by the above rule as $x$ and $y$ are coprime.

My questions:

1) What is $J$ in the result?

2) In the example which I have given, who are $a,b$ and $J$?

  • I have seen this in math. Stackexchange which I don't understand that is why I posted it.

Thanks

Jaca
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Pradip
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    Where have you seen it at MSE? I saw this post, which is a bit different. – Dietrich Burde Feb 20 '20 at 10:44
  • I have seen in another. I read your link now. But stumble because I don't know the term " Pure power" . Can you tell me this please – Pradip Feb 20 '20 at 11:21
  • A pure power of the variable $x$ is given by $x^k$ for some positive integer $k$. If you have two variables $x,y$ then $x^k$ and $y^j$ are pure powers, but $xy$ is not. What is your "another link" at mathematics stack exchange? – Dietrich Burde Feb 20 '20 at 20:13
  • Then what is the difference between power and pure power? I don't know how to share a link in MSE, Sorry – Pradip Feb 21 '20 at 02:47
  • $(xy)^2$ is a power in the variables $x,y$ but not a pure power? – Dietrich Burde Feb 21 '20 at 08:48

1 Answers1

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1) In the Result $J$ denotes an ideal generated by $a^n$ or by $b^n$ for some positive integer $n$.

2) In your example $a=x,b=y$ and $J=(y^2)$, i.e, $J$ is the ideal generated by $y^2$. Thus, since $I=(xy,y^2)=(xy)+(y^2)$, by your Result you have

$$I=(xy,y^2)=(xy)+(y^2)=(\overbrace{(x)+(y^2)}^{(x,y^2)})\cap(\overbrace{(y)+(y^2)}^{(y,y^2)=(y)}).$$

Jaca
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