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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