# Determining if ___ is an ideal:

I am reading some notes on algebra, with polynomial rings and there is an exercise that asks to determine if 1-5 are ideals. I am not very familiar and am hoping that the examples will give a more concrete understanding of what an ideal is (I understand the definition). I would appreciate if someone would answer with a reason why as well (though it does not need to be a full proof).

(1) k, the field of coefficients;

(2) a subring $k[x_1,...,x_m] \subset R = k[x_1,...,x_n]$, where $0 < m < n$;

(3) polynomials with no constant term;

(4) $R\leq d$, polynomials of degree at most $d$;

(this one would not be a subring, say $S$, because $x^d \in S$ but $x^d*x^d \not \in S$)?

(5) homogeneous polynomials, i.e., polynomials with all terms of the same degree.

(as with 4, this would not be closed under multiplication)?

Thank you!

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Only one of them is actually an ideal, so perhaps start looking for reasons why they are not ideals and we can check that. –  Karl Kronenfeld Feb 12 '14 at 23:54

(1) $k$ is not an ideal, since it is not closed under multiplication using elements from outside $k$. For instance $1x_1\not\in k$.
(5) You are correct, it is not closed under multiplication by outside elements. To be clear, we are multiplying by something like $x+1$. Also not closed under addition if the degrees vary.