Scalars, tensor powers, the empty tensor, opposite algebras and finite sums

Fix the following conventions, used on this page and by the later Hopf and Hecke pages.

• $k$ is a field (Field) and every vector space is a $k$-vector space (Vector space over a field); $\otimes$ means ${\otimes }_k$.
• $V\otimes W$ is the tensor product of The tensor product $M{\otimes }_{R}N$ from the additive group underlying the free $\mathbb{Z}$-module on $M\times N$, elementary tensors, and finite tensor sums, with the $k$-vector-space structure of Over a commutative ring, $M{\otimes }_{R}N$ is an $R$-module with $r(m\otimes n)=(rm)\otimes n=m\otimes (rn)$ and unit and universal property as in Universal property of the tensor product for balanced maps into abelian groups. Every element of $V\otimes W$ is a finite sum of elementary tensors $v\otimes w$.
• Tensor powers are left-associated: $V^{\otimes 0}:=k$, $V^{\otimes 1}:=V$ and $V^{\otimes n}:=(V^{\otimes (n-1)})\otimes V$ for $n\ge 2$. The empty tensor $k$ is identified with the tensor unit through the isomorphisms of The regular module is a tensor unit: $R{\otimes }_{R}N\cong N$ and $M{\otimes }_{R}R\cong M$.
• For an algebra $A$ over a commutative ring (Algebras over a commutative ring, central structure maps, and algebra homomorphisms) the opposite algebra $A^{\mathrm{o}\mathrm{p}}$ has the same underlying set and structure map and product $a\cdot b:=ba$ (The opposite ring $R^{\mathrm{o}\mathrm{p}}$).
• Finite sums ${\sum }_{i\in I}{x}_i$ follow A finite sum in a commutative monoid indexed by an arbitrary finite set; an empty sum is $0$.

Parentheses in iterated tensor products may be dropped only after Associator naturality, pentagon, unit triangle and symmetry hexagons on elementary tensors ↗ has been proved.