Definition 1

For every module $M$ over a commutative ring $A$, the tensor algebra of $M$, denoted by $T(M)$, or $\mathrm{Tens}(M)$, or $T_A(M)$, is the algebra $\bigoplus_{n \geq 0} T^n(M)$ with the multiplication defined in (1). The canonical injection $\phi : T^1(M) \to T(M)$ (II, § 1, no. 12) (also denoted by $\phi_M$) is called the canonical injection of $M$ into $T(M)$.