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