I have come up with the following two different decompositions of the ideal $I= (X^2, XY)$: $I = (X) \cap (X^2, Y)$ and $I = (X) \cap (X^2, XY, Y^2) = (X) \cap (X, Y)^2$. Can we generalize this somehow so that there are are infinitely many different primary decompositions?

For an ideal $I \subseteq k[X_1, \dots ,X_n]$ prove that the following are equivalent: $I$ is generated by monomials. If $f =\sum \limits _a c_a X^a \in I$, and $c_a \ne 0$, then $X^a \in I$, where $a=(a_1, \dots ,a_n) \in \mathbb Z _+ ^n$.

Show that $I=(X_{k_1}^{a_1},…,X_{k_s}^{a_s})$ is $(X_{k_1},…,X_{k_s})$-primary. I noticed that $$\sqrt{({X_{i_1}}^{a_1},…,{X_{i_k}}^{a_k})}=\sqrt{({X_{i_1}}^{a_1})+\cdots+({X_{i_k}}^{a_k})}=\sqrt{\sqrt{(X_{i_1})^{a_1}}+\cdots+\sqrt{(X_{i_k})^{a_k}}}=\sqrt{(X_{i_1})+\cdots+(X_{i_k})}=\sqrt{(X_{i_1},…,X_{i_k})}=(X_{i_1},…,X_{i_k}).$$ I have to show that $I$ is $P$-primary (where $P=(X_{k_1},…,X_{k_s})$), i.e., if $u v\in I$ then $u\in I$ or $v\in P$. I tried to prove that if $u\notin I$ and $v\notin P$ then $uv\notin I$. (There is a similar question, but the answer is too hard […]

I need to show that given a term order $<$, and an ideal $I$, if $\mathrm{in}_<(I)$ is radical, then $I$ is radical. Any help or hints would be appreciated as I’m not really sure where to start, since I know a few different facts about radical ideals.

I have problems with this: I need to prove that in the polynomial ring the radical of an ideal generated by monomials is also generated by monomials. I found a proof on internet that uses the convex hull of the multidegrees of the monomials, but I want a proof that uses less terminology. For example, […]

Can $(X_1,X_2) \cap (X_3,X_4)$ be generated with two elements in the ring $R=k[X_1,X_2,X_3,X_4]$? Can it be generated with three elements? (Here $k$ is a field.) Thanks for any help.

Can anyone give me an idea about the primary decomposition of the ideal $I=(x^3y,xy^4)$ of the ring $R=k[x,y]$? I am trying to connect the primary decomposition with the set Ass(R/I) which i have find before..

I want to find a primary decomposition of the ideal $$ I = (x^2,y^2,xy) \subset k[x,y]$$ where $k$ is a field. How to proceed? Are there algorithms to find such decompositions? Where can I find them?

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