[Math] How to construct polynomial ring $K[x]$ over commutative ring $K$ by making use of universal arrows.


In CWM of Mac Lane I encounter:

the construction of a polynomial ring
$K\left[x\right]$ in an indeterminate $x$ over a commutative ring
$K$ is a universal construction.

Unfortunately this as an exercise (on pg 59) and I don't manage to
solve it. Can someone help me with this?

Best Answer

The universal property of the polynomial ring $K[x]$ is that

$$\hom_K(K[x], R) \simeq |R|,$$

where $\hom$ is taken in the category of $K$-algebras, and $|R|$ is the underlying set of $R$. The bijection is determined by looking at the image of the free variable $x$. In other words, a free variable is free to go where it wants.

Another way to say this is that $K[x]$ represents the forgetful functor from $K$-algebras to sets.

In MacLane's language, the element $x \in |K[x]|$ is the "universal element" for the forgetful functor.

Related Question