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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
up vote 3 down vote accepted

Considering that the list of justifications hasn't been fully populated, I will provide an answer so as to remove this from the unanswered queue.

(1) $k$ is not an ideal, since it is not closed under multiplication using elements from outside $k$. For instance $1x_1\not\in k$.
(2) This is basically the same as (1).
(3) This is actually an ideal!
(4) You are correct.
(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.

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