Why are quotients of polynomial rings called "affine algebras" in Magma?

Question

Although quotients of polynomials rings are affine algebras (in the sense that they are coordinate rings of affine schemes), they are not the only ones, right?

Answer 1

The term affine algebra (over $k$) usually refers to finitely generated $k$-algebras; that is, quotients of finite variable polynomial rings over $k$. (In the commutative case; in the non-commutative case just take the tensor algebra instead.) This is motivated not by affine schemes (over $k$)—they indeed correspond to all $k$-algebras—but from the fact that the quotients of $k[x_1,\ldots,x_n]$ are the coordinate rings of closed sub-schemes of $\mathbb{A}^n_k$. (Up to possible non-irreducibility or non-reducedness, these are the scheme versions of affine varieties over $k$.)