How do I prove that this ideal is not a ' ideal?Let K be a field and&nbsp;R=K[X,Y](XY).F&nbsp;or&nbsp;P∈K[X,Y]&nbsp;we denote&nbsp;[P]&nbsp;its class in R. Show that the Ideal (XY) is not a ' ideal.

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How do I prove that this ideal is not a &#039; ideal?Let K be a field and&amp;nbsp;R=K[X,Y](XY).F&amp;nbsp;or&amp;nbsp;P∈K[X,Y]&amp;nbsp;we denote&amp;nbsp;[P]&amp;nbsp;its class in R. Show that the Ideal (XY) is not a &#039; ideal.

Alonso Christian · Accepted Answer

Step 1Here is one possible proof:Let&amp;nbsp;φ:K[X,Y]→Kbe the unique K-algebra homomorphism such that&amp;nbsp;φ(X)=1&amp;nbsp;adn&amp;nbsp;φ(Y)=0. Thenφ(XY)=φ(X)φ(Y)=1⋅0=0&amp;nbsp;so this factors through the quotientK[X,Y]→K[X,Y](XY)=R&amp;nbsp;to give a homomorphismφ~:R→K&amp;nbsp;such that&amp;nbsp;widetilde{φ}([P])=φ(P)&amp;nbsp;for all&amp;nbsp;P∈K[X,Y]Since K is a field,&amp;nbsp;φ~([X])=φ(X)=1≠, so&amp;nbsp;[X]≠0In other words,&amp;nbsp;Y∉(XY)Symmetrically, we must have&amp;nbsp;X∉(XY), showing that&amp;nbsp;(XY)&amp;nbsp;is not &#039;.

How do I prove that this ideal is

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