As I said before, if we take the free commutative monoid on generators, then build the semigroup ring from that, the result is the ring of polynomials in variables. I hinted at a noncommutative analogue, which today I’ll construct from the other side.
Instead of starting with a set of generators and getting a monoid, let’s start by building the free abelian group . This consists of ordered -tuples of integers, and we add them component by component. We can pick out the generators , where the shows up in slot . Then every element can be written , where the is entry in the -tuple form of the element.
So how do we build the tensor product ? First we take all pairs
and use them to generate a free abelian group. Then we impose the linearity relations and . What does that mean here? Well for one thing we can apply it to the collection of pairs:
So we could just as well write the tensor product as the group generated by .
This same argument goes through as we tensor in more and more copies of . The tensor power is the free abelian group generated by the elements , where each index runs from to .
Now we take all of these tensor powers and throw them together. We get formal linear combinations
where all but finitely many of the “coefficients” are zero. These look an awful lot like polynomials, don’t they? In fact, if we only had a commutative property that then these would be exactly (isomorphic to) the polynomials we came up with last time.
To be explicit about the universal properties, any function from the generators to the underlying abelian group of a ring with unit has an unique extension to a linear function from to . Then this has a unique extension to a ring homomorphism from to . From the other side, there is a unique extension of the original function to a monoid homomorphism from the free monoid to the underlying monoid of . Then this has a unique extension to a ring homomorphism from to . Since both and satisfy this same universal property they must be isomorphic. We commonly write this universal ring as , and call it the ring of noncommutative polynomials in variables.