In Hatcher's book, on page 227, he said
Polynomial algebras are examples of free graded commutative algebras, where ‘free’ means loosely ‘having no unnecessary relations.’ In general, a free graded com- mutative algebra is a tensor product of single-generator free graded commutative algebras. The latter are either polynomial algebras $R[α]$ on even-dimension generators α or quotients $R[α]/(2α^2)$ with α odd-dimensional. Note that if $R$ is a field then $R[α]/(2α^2)$ is either the exterior algebra $Λ_R[α]$ if the characteristic of $R$ is not 2, or the polynomial algebra $R[α]$ otherwise. Every graded commutative algebra is a quotient of a free one, clearly.
My question is:
Why is $R[\alpha]/2\alpha^2$ a free graded commutative algebra? I can understand that when $R$ is a field and $R[\alpha]/2\alpha^2$ is either a exterior algebra or a polynomial algebra. Therefore, $R[\alpha]/2\alpha^2$ is a free graded commutative algebra in this case. However, if $R$ is not a field, can we say it is a free graded commutative algebra?