Which of the following subsets of $R[x]$ are subrings of $R[x]$? Prove or disprove.
All polynomials with constant term $0_R$.
The elements of this set have the form $r_nx^n + r_{n-1}x^{(n-1)} + \dots + r_{1}x$. This is closed, since $R[x]$ does not contain zero divisors.
All polynomials of degree $2$.
This is the subset $F$ with the elements of form $a_2n^2 + a_1n + a_0$. It is clear it is closed under subtraction. It is not closed under multiplication, for multiplying the first two terms of any elements of this set together yields $(a_2n^2)(b_2n^2) = a_2b_2n^4$, a polynomial not in the set $F$.
All polynomials of degree $\leq k$ where $k$ is a fixed positive integer.
Consider the set $G$ of polynomials with $k = 2$. Then there are two polynomials with degree $2$ that yield, under multiplication, an element not in $G$.
All polynomials in which the odd powers of $x$ have zero coefficients. (Consider the even powers as well.)
Call the set of polynomials in which the odd powers of $x$ have zero coefficients $H$. Then $H$ has elements of the form $h_0 + h_2x^2 + h_4x^4 + \dots + h_{2n}x^{2n}$. It is clearly closed under subtraction. Since an even number added to an even number yields an even number, the set $H$ is a subring of $R[x]$.
Call the set of polynomials in which the even powers of $x$ have zero coefficients $J$. Then $J$ has elements of the form $h_1x + h_3x^3 + \dots + h_{2n-1}x^{2n-1}$. It is clearly closed under subtraction. Since an odd number added to an odd number yields an even number, the set $J$ is NOT a subring of $R[x]$.