## Polynomial rings

§1 Introduction. Today, we will let $R$ be a commutative ring with a multiplicative identity (we usually assume it to at least be an integral domain when talking about its polynomial ring). In this post I want to talk about the polynomial ring $R[x]$ and attempt to explain, from the fairly abstract standpoint of universal properties, the reasons for its general ubiquity throughout the realm of algebra.

§2 The notion of an algebra. In order to really flesh out what makes $R[x]$ so important, it is necessary to define something known as an $R$-algebra. There are many not-so-equivalent definitions of this notion depending on, for example, whether you require multiplication to be associative or not. Henceforth by algebra I mean “associative commutative unital algebra”, however we will not restrict ourselves to $k$-algebras for some field $k$. This translates into saying that the underlying “linear structure” of this thing need not be that of a vector space, but merely that of a module. The general idea is that to call something an $R$-algebra, we want it to have an “addition”, a “multiplication”, and a “scalar multiplication by elements of $R$” similar to what is familiar from linear algebra (or more generally, module theory). Which one of these is the “extra” operation depends on which way you look at it. Remark that for a ring $R$ as defined above, the ring multiplication is always associative. Therefore we could define an (associative, commutative, unital) $R$-algebra to be, well, a commutative unital ring $S$ together with a scalar multiplication map, that is, an action of $R$ on $S$, in such a way that the ring multiplication becomes $R$-bilinear. At this point, since $S$ is unital, we can identify $R$ with the subring

$\{ r \cdot 1 : r \in R \}$,

where $1 \in S$ is the identity and $\cdot$ is the scalar multiplication. Alternatively, we can define an $R$-algebra to be a commutative unital ring $S$ together with a ring homomorphism $\varphi : R \to S$ (here, ring homomorphisms are required to map identity to identity!). We then identify $R$ with its image under this map $\varphi$. As a third alternative, instead of “starting with a ring and adding the scalar multiplication”, we can also start with a module and add the ring multiplication. Then we say an $R$-algebra is an $R$-module $S$ together with a vector multiplication on $S$, such that the induced thing $(S,+,\cdot)$ turns into a commutative, unital ring. It is interesting to note that, for example, when we consider vector spaces, they are basically just abelian groups under addition, being acted on by a field. For this reason, linear algebra only “sees” the 0 element in the underlying set because the set is not even thought of as having any multiplicative structure; it’s simply an additive group.

Having defined an $R$-algebra it is natural to ask about morphisms between such objects, so that hopefully the set of $R$-algebras together with the correct notion of “morphism” will form a category. Suppose $S, T$ are $R$-algebras, such that $\varphi_S : R \to S$ and $\varphi_T : R \to T$ give us the homomorphisms of $R$ into those rings, as discussed above. We say that a unital ring homomorphism $\psi : S \to T$ is a homomorphism of $R$-algebras if it “preserves the scalars”, in other words $\psi(\varphi_S(a)) = \varphi_T(a)$ for any $a \in R$, or more briefly, $\psi \circ \varphi_S = \varphi_T$. This is really a statement about a certain diagram being commutative.

§3 Formal development of polynomials. Being familiar with polynomials from years of experience in grammar school, it is tempting to simply define a single-variable polynomial as an “expression” of the form $a_0 + a_1x + \ldots + a_nx^n$ where $a_i \in R$ and $n$ is some natural number. However, this is hardly a sufficient level of precision. Instead, we proceed as follows: consider the set of all sequences $(a_1, a_2, \ldots)$ where each $a_i \in R$. This set is often written

$\prod\limits_{n \in \mathbb{N}} R$

for the “Cartesian product of countably many copies of $R$“. Note that $\mathbb{N}$‘s property of being well-ordered gifts us with the wonderful ability of representing the elements of the above set as ordered sequences (they are actually functions $\mathbb{N} \to R$, of course). We now look at the subset of this set consisting of all sequences $(a_1, a_2, \ldots)$ which are “ultimately zero”, that is such that there is $\ell \in \mathbb{N}$ with $a_n = 0_R$ for all $n \geq \ell$. We define algebraic operations on these sequences in such a way to “mimic” the usual operations on polynomials, and then we define $x=(0,1,0,\ldots)$, the sequence with a 1 in only the second position. Observe that $x^n = (0,0,\ldots,1,0,\ldots) = \delta_{i,n+1}$ (Kronecker delta notation). It is easily checked that this thing becomes a commutative unital ring. We call this the ring of polynomials in one variable $R[x]$ and identify $R$ with a subring of it, by virtue of the injective ring homomorphism $a \mapsto (a,0,\ldots)$. That is, we turn $R[x]$ into an $R$-algebra.

§4 The “most general” way of adjoining an element. What makes the polynomial ring so special? Observe that in our formal construction, we made sure that the elements $x^n$ ($n \in \mathbb{N}$) were all considered “distinct” in the ring $R[x]$. This is important. It means that our “indeterminate” $x$ satisfies no additional relations other than the ones imposed by ring axioms. This implies that the ring $R[x]$ is an extremely “general” object. It’s a way of forming a new ring, which contains $R$ in addition to a new element $x$ which “we know absolutely nothing about”. If we take $R=\mathbb{Z}$, view it as a subring of $\mathbb{C}$ and consider the ring of Gaussian integers,

$\mathbb{Z}[i] = \{ a_0 + a_1i + a_2i^2 + \ldots : a_i \in \mathbb{Z} \}$

then we see, of course, by the cyclic behaviour of the powers of $i$, that indeed every element here has the form $a + bi$ for some $a,b \in \mathbb{Z}$. So indeed, this thing kind of “looks like” a polynomial ring, except that we know stuff about the adjoined element. In particular, $i$ satisfies the equation $i^2 + 1 = 0$. It therefore comes as no surprise that $\mathbb{Z}[i]$ is NOT isomorphic to the ring of integer-coefficient polynomials in one variable, but rather a quotient of that ring by the ideal $\langle x^2 + 1 \rangle$. We are starting with something free and then quotienting out to impose the desired relations.

On the other hand, consider now $R = \mathbb{Q}$ and view this as a subring of the real numbers $\mathbb{R}$. If we take the element $e \in \mathbb{R}$ and think about the smallest subring of $\mathbb{R}$ containing $\mathbb{Q}$ and $e$, well, it turns out that we actually get $\mathbb{Q}[e] \cong \mathbb{Q}[x]$. If you’ve taken field theory this is no surprise; $e$ is transcendental over the rationals, so it behaves in such an unruly way when adjoined that it “might as well just be an indeterminate”.

§5 The universal property of the polynomial ring. Well, unfortunately it’s time for me to finish off this post, so without further ado, here we go. What is really so “special” about the ring $R[x]$? Well, it satisfies the following property:

For any $R$-algebra $S$, and element $\alpha \in S$, there is a unique homomorphism of $R$-algebras $\psi : R[x] \to S$ such that $\psi(x) = \alpha$.

This is often expressed by saying that the polynomial ring is a free commutative $R$-algebra. I will not elaborate on this much, but freeness is related to the notion of adjoint functors. Think about what this means for several concrete examples. Which part of this property fails if I run out into the street and claim that $\mathbb{Z}[i]$ is the polynomial ring of $\mathbb{Z}$?

Thinking about these kind of properties can be used to gain deep insight into why certain notions in mathematics are “worth considering”, although this is certainly not at all what makes category theory and related notions so powerful. A lot of indispensable objects can be characterized completely by universal mapping properties such as the one I described above; the polynomial ring $R[x]$ is just one example. Tensor products give another example. Projection maps onto quotient groups give yet another example. Start pondering this question to yourself whenever you see a new algebraic definition: what makes this special? Maybe you can even come up with the correct universal property yourself, although it is often a subtle matter.