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Abstract Algebra

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Sets that are Ideals in Rings of Polynomials

ABSALG-LAKYQO

Recall that an ideal ${\mathcal I}$ in a commutative ring $R$ is a subring of $R$ such that for all $r \in R$ and all $\iota \in{\mathcal I}$, we have $r\iota\in{\mathcal I}$.

Let $k$ be a field and $R=k[x,y,z]$ be the ring of polynomials in $3$ variables with coefficients in $k$.

Let $k^3$ be the set of ordered triples $(a,b,c)$ with $a,b,c\in k$.

Which of the following subsets of $R$ forms an ideal in $R$?

Select ALL that apply.​

A

The polynomials vanishing at the points of a non-empty subset $S$ of $k^3$

B

The polynomials vanishing on either $S$ or $T$ but not both, where $S$ and $T$ are non-empty distinct subsets of $k^3$

C

The polynomials $P$ such that $P^n\in {\mathcal I}$, for some integer $n\ge 1$, where ${\mathcal I}$ is a fixed ideal in $R$

D

The constant polynomials

E

The polynomials with no constant term