# Rings of Polynomials

This page is a sub-page of our page on Some basic algebraic concepts.

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Formalizing the notion of direct sum

Let $X$ be an arbitrary set and let $R$ be a ring with unit element. If ${f : X \, \rightarrow \, R}$, we define the support $\mathrm{supp} f$ of the function $f$ as

$\mathrm{supp} f \, \stackrel {\mathrm{def}}{=} \{ x \in X \, : \, f(x) \neq 0 \} \, .$

Definition: The direct sum of $R$ over $X$ is the set of all functions from $X$ to $R$ with finite support. It is denoted by ${\bigoplus \atop { X } } R$ or ${\coprod \atop { X } } R$.

Let $f, g \in {\bigoplus \atop { X } } R$ and $r \in R$. If we define

$(f+g)(x) \stackrel {\mathrm{def}}{=} f(x) + g(x) \, , \text{ and} \, (rf)(x) \stackrel {\mathrm{def}}{=} r(f(x)) \, ,$

we turn ${\bigoplus \atop { X } } R$ into an $R$ -module. If $R$ is a field, then these definitions turn ${\bigoplus \atop { X } } R$ into a vector space over $R.$

If $x \in X$ we can define the characteristic function $\hat{x}$ of the element $x$ as the function ${\hat{x} : X \, \rightarrow \, R}$, which takes the values $\hat{x}(x) = 1$ and $\hat{x}( y) = 0$ if $y \neq x.$ Using this definition, we can write a function $f \in {\bigoplus \atop { X } } R$ as

$f \equiv {\sum\limits_{x \in X}^{ \text {} }}f(x) \hat{x}$

where, since $\mathrm{supp} f$ is finite, the sum only contains a finite number of terms.

Notation: This sum is most often written as

$f \equiv {\sum\limits_{x \in X}^{ \text {} }}f(x) x \, , \, \text {or} \, f \equiv {\sum\limits_{x \in X}^{ \text {} }}{f_x} x$

and it is interpreted as a formal (or direct) sum of elements in $X$ with coefficients in $R$. Hence, a more “semantically precise” way of writing this direct sum (which is often used in abstract algebra) is

$f \equiv {\bigoplus\limits_{x \in X}^{ \text {} }}f(x) x \, .$

The $R$ module of Lists (of elements from an arbitrary set)

Let $X$ be an arbitrary set and define the cartesian products of $X$ inductively by $X^0 \stackrel {\mathrm{def}}{=} \emptyset$ , $X^1 \stackrel {\mathrm{def}}{=} X$ , $X^{n+1} \stackrel {\mathrm{def}}{=} X \times X^n.$ The set $List(X)$ of all finite lists of elements of $x \in X$ is defined by

$List(X) \stackrel {\mathrm{def}}{=} \bigcup\limits_{n \geq 0}^{ \text {}} X^n \, .$

Let ${\mathrm{concat} : List(X) \times List(X) \, \rightarrow \, List(X)}$ denote concatenation of lists, i.e.,

${List(X) \times List(X) \ni (x_1 x_2 \ldots x_m, y_1 y_2 \ldots y_n ) \, \mapsto \, x_1 x_2 \ldots x_m y_1 y_2 \ldots y_n \in List(X)} \, .$

The operation $\mathrm{concat}$ is a binary composition on $List(X)$ that we will refer to as multiplication of lists. $\mathrm{concat}$ is obviously associative, and the empty list $\emptyset,$ which we can call $1,$ works as a unit element under $\mathrm{concat}.$ Hence, with these definitions, $(List(X), \mathrm{concat})$ has the structure of a monoid, i.e., a semigroup with a unit element.

Terminology: The elements of $List(X)$ are called monomials.

The non-commutative polynomial ring in the variables $X$ over the ring $R$

Let $M = List(X)$ and let $R$ be a ring with unit element. We define

$R\{X\} \stackrel {\mathrm{def}}{=} {\bigoplus\limits_{M}^{ \text {} }}R \, .$

Terminology: An element $p \in R\{X\}$ is called a polynomial with variables from $X$ and coefficients from $R.$

We can write $p$ as a formal sum of “scaled monomials”, i.e., as a sum of monomials $m \in M$ with corresponding coefficients $p_m \in R:$

$p = {\sum\limits_{m \in M}^{ \text {} }}{p_m} m \, ,$

Terminology: Each scaled monomial ${p_m} m$ is called a term of the polynomial $p.$

We will now define a multiplication $\ast$ on $R\{X\}$ which extends the multiplication $\mathrm{concat}$ on $M$ so that $(R\{X\}, \ast)$ becomes a ring.

Definition: The multiplication ${\ast : R\{X\} \times R\{X\} \, \rightarrow \, R\{X\}}$ is defined by:

$R\{X\} \times R\{X\} \ni (f, g) \, \mapsto \, f \ast g \in {R\{X\}\, , \text{where} \, (f \ast g)(m) \stackrel {\mathrm{def}}{=}{\sum\limits_{m'm'' = m}^{ \text {} }f(m')g(m'')}}$

Hence, when we compute $f \ast g,$ we sum over each pair of monomials $(m', m'')$ which satisfies the conditions $m' \in \mathrm{supp} f \, , m'' \in \mathrm{supp} g,$ and $m'm'' = m.$

Notation: The multiplication $\ast$ is most often expressed by concatenation of the factors, i.e., $fg \,\stackrel {\mathrm{def}}{=} \, f \ast g.$ Moreover, if we regard $f$ and $g$ as formal sums, i.e.,

$f = {\sum\limits_{m \in M}^{ \text {} }}{f_m} m \, , \text{ and } \, g = {\sum\limits_{m \in m}^{ \text {} }}{g_m} m \, ,$

we get

$fg = {\sum\limits_{m \in M}^{ \text {} }}f_{m'} g_{m''}m'm'' \, .$

Here each monomial $m \in M$ corresponds to a sum of all of its possible “factor monomials” $m', m'',$ which satisfy $m'm'' = m,$ i.e., $m',$ and $m'',$ concatenate to $m.$