Definition 1

Let A be a commutative ring and M an A-module. The symmetric algebra of M, denoted by S(M), or Sym(M), or SA(M), is the quotient algebra over A of the tensor algebra T(M) by the two-sided ideal $\mathfrak{J}'$ (also denoted by $\mathfrak{J}'_M$) generated by the elements $xy - yx = x \otimes y - y \otimes x$ of T(M), where x and y run through M.*

Since the ideal $\mathfrak{J}'$ is generated by homogeneous elements of degree 2, it is a graded ideal (II, § 11, no. 3, Proposition 2); we write $\mathfrak{J}'_n = \mathfrak{J}' \cap T^n(M)$; the algebra S(M) is then graded by the graduation (called canonical) consisting of the $S^n(M) = T^n(M)/\mathfrak{J}'_n$. Now $\mathfrak{J}'_0 = \mathfrak{J}'_1 = \{0\}$ and hence $S^0(M)$ is canonically identified with A and $S^1(M)$ with $T^1(M) = M$; we shall always make these identifications in what follows and denote by $\phi'$ or $\phi'_M$ the canonical injection $M \to S(M)$.