Im having trouble with an exercise. Am I missing some information on the ring $R$?
Let $R$ be a ring. Which of the following are subrings of $R[X]$
Polynomials of degree less than $5$
Polynomials in which the odd powers of $X$ have coefficient $0$
Polynomials in which the even powers of $X$ have coefficient $0$
If you consider $x^5 \cdot x^4=x^9$ you have shown that the polynomials of degree less than $5$ do not form a subring. Also for the even powers with coefficient $0$ a similar counterexample is $x^3 \cdot x^3=x^6$; check by yourself if the other set is a subring!