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)?