Standard definition of a Tensor Algebra?

Question

Let $M$ be a $R$-module. I know that the set of a tensor algebra is given by:

$$T(M)=\bigoplus_{k=0}^\infty T^k(M)$$

where $T^K(M)=M\otimes ...\otimes M$ ($k$ times). But I am confused about the fundamental operations defined along with this set (see the definition of an algebra given here (pg2) ). The three possible onces are:

1. Addition of tensors of the same rank.
2. Tensor product.
3. Multiplication by a scalar.

Different sources seem to include different combinations of these when defining a tensor algebra (many don't even mention the operators). My question is; which operators do we, as standard, take when defining the tensor algebra? (sources would be helpful).

Answer 1

Like the name suggests, $T(M)$ should be a (graded, associative) $R$-algebra (with unit). This means one should describe quite a few structures on $T(M)$:

1. The set $T(M)$ should have the structure of an $R$-module. Hence, there should be an addition map $T(M) \times T(M) \rightarrow T(M)$ and a scalar multiplication map $R \times T(M) \rightarrow T(M)$ which are compatible with each other according to the axioms of an $R$-module. The addition should be defined between any two tensors, not only tensors of the same rank (or even homogeneous tensors).
2. The set $T(M) = \bigoplus_{k=0}^{\infty} T^k(M)$ has a grading given by the direct sum.
3. The set $T(M)$ should have an $R$-bilinear multiplication operation $T(M) \times T(M) \rightarrow T(M)$ which gives it the structure of an $R$-algebra. This multiplication is associative, has a unit and respects the grading. On homogeneous and elementary tensors, it is given by $$(m_1 \otimes \cdots \otimes m_k) \cdot (w_1 \otimes \cdots \otimes w_l) = m_1 \otimes \cdots \otimes m_k \otimes w_1 \otimes \cdots \otimes w_l. $$

This is quite tedious to do in full detail so often enough textbooks assume the reader is mature enough to fill in the details behind all those structures.