Product of elements in tensor algebra

Question

I am having trouble understanding a part of a proof given in Dummit and Foote relating to tensor algebras. The theorem states: Suppose $M$ is any $R$ module where $R$ is a commutative ring with $1$. Then $\mathcal{T}(M)$ is an $R$-algebra with multiplication defined by the mapping $(m_1\otimes\cdots\otimes m_i)(m_1'\otimes\cdots\otimes m_j')=m_1\otimes\cdots\otimes m_i\otimes m_1\otimes\cdots\otimes m_j'$ and extended to sums via distributive law. With respect to this multiplication $\mathcal{T}^i(M)\mathcal{T}^j(M)\subseteq\mathcal{T}^{i+j}(M)$.

The proof starts as follows:The map $M\times\cdots\times M\times M\times\cdots \times M\rightarrow\mathcal{T}^{i+j}(M)$ defined by $(m_1,...,m_i,m_1',...,m_j')\mapsto m_1\otimes\cdots\otimes m_i\otimes m_1'\otimes\cdots\otimes m_j'$ is $R$-multilinear, so induces a bilinear map $\mathcal{T}^i(M)\times\mathcal{T}^j(M)$ to $\mathcal{T}^{i+j}(M)$.

Why does the multilinear map above induce this bilinear map? Does this somehow follow from the universal property for tensor products? I know in general multilinear maps induce homomorphisms from tensor products, but that doesn't seem to be what's going on here.

Answer 1

Good question. It's not so clear, and there's actually some work to be done. However, the work is similar to the work done in Proposition 10.21: namely, the work is that Corollary 10.16 is used twice, and both uses are for only the part of proving that the ring multiplication is well-defined. I guess Dummit Foote assume the readers of Chapter 11.5 (specifically of Theorem 11.31) have also read Chapter 10.4 (specifically of Proposition 10.21 and Corollary 10.16).

(and then some parts with errors. see the official errata, 'most recently revised on February 14, 2018' or 'most recently revised on March 17, 2019')

Answer 2

You have to check that $\varphi \colon T^i(M) \times T^j \rightarrow T^{i+j}(M), \big((m_1 \otimes \ldots \otimes m_i) ,(m^\prime_1 \otimes \ldots \otimes m^\prime_j)\big) \mapsto m_1 \otimes \ldots \otimes m_i \otimes m^\prime_1 \otimes \ldots \otimes m^\prime_j \\$ is bilinear. For this you can use the R-multi lin. of the given map.