Question

Are polynomials closed under sub

Answer 1

traction?

Answer 2

yes
take r in R[x] of degree n
r=a_n x^n+...+a0
r'=b_n x^m+...+b0
WLOG
m>n
r'-r=(b_m-0)x^m+...+(b_n-a_n)x^n+...+(b_n-a_n)
which is in R[x]