This article will motivate the example of a ring, a common algebraic structure, using various examples. Briefly, we basically want to see what happens when we add a second operation to a group.

Integers

This is the canconical ring in some sense since the integers $\mathbb{Z}$ are typically associated with two operations $+$ and $\times$. In addition, $(\mathbb{Z}, +)$ is a commutative group, while $(\mathbb{Z},\times)$ doesn’t even have inverses! However, these two operations are also compatible with each other via the distribitivity axioms, $\forall a, b, c \in \mathbb{Z}, a(b+c) = ab + ac$ and $(a+b)c = ac + bc$. Somewhat interestingly, this is a common structure.

Square Matrices

In this case, we consider the space of $n \times n$ matrices over $\mathbf{R}$, for instance. We denote this space as $M_{n\times n(\mathbb{R}})$. Adding matrices is simply defined as \(\begin{align} A + B &= [ a_{ij} ]_{n \times n} + [ b_{ij} ]_{n \times n} =\left[ a_{ij} + b_{ij} \right]_{n \times n} \end{align}\)

i.e. it is element-wise addition. Clearly, we immediately get associativity as \(\begin{align} (A + B)+C &= ( \left[ a_{ij} \right]_{n \times n} + \left[ b_{ij} \right]_{n \times n} ) + \left[ c_{ij} \right]_{n \times n} \nonumber \\ &= \left[ a_{ij} + b_{ij} \right] + \left[ c_{ij} \right] \nonumber \\ &= \left[ a_{ij} + b_{ij} + c_{ij} \right] \nonumber \\ &= \left[ a_{ij} + (b_{ij} + c_{ij}) \right] \nonumber \\ &= \left[ a_{ij} \right] + \left[ b_{ij} + c_{ij} \right] \nonumber \\ &= A + (B+C) \end{align}\)

where we used the associativity of $\mathbb{R}$ and the definition of the operation. Similarly, we can show that the zero matrix $\mathbf{0}$ is the additive identity, and for each matrix $A$, $-A$ is the additive inverse. It is also trivially commutative under addition as well.

For the multiplication operation, we consider regular matrix multiplication. It admits an multiplicative identity $\mathbf{I}$, but we cannot guarantee that each matrix has an inverse. We can quickly show distributivity in Einstein notation as \(\begin{align} D = A(B+C) &\implies d_{ik} = a_{ij} (b_{jk} + c_{jk}) = a_{ij}b_{jk} + a_{ij}c_{jk} = AB + AC \end{align}\)

as desired. Note however that this does not really tell us anything interesting, and proofs viewing matrices as linear transformations might be worth a look.

In short, the space of square matrices is a ring under element-wise addition and matrix multiplication.

Continuous Functions

Finally, we consider the space of all continuous real-valued functions, $\mathcal{C}(\mathbb{R}) = { f: \mathbb{R} \rightarrow \mathbb{R} }$. Under pointwise addition, i.e. $(f+g)(x) = f(x) + g(x)$, we can easily see that it forms a commutative group. The constant function $f_0(x) = 0, \forall x$, is the additive identity, and the constant function $f_1(x) = 1, \forall x$, is the multiplicative identity. Distributivity is also quite natural to show, since we always have for example, $f(x) (g(x) + h(x)) = f(x)g(x) + f(x)h(x)$, for any three such functions $f,g,h$. However, the inverse of a function is not necessarily defined, since any function with a zero i.e. $\exists x \in \mathbb{R}$ such that $f(x) = 0$, cannot have a (pointwise) inverse.

General Definition

To talk about all these examples at once, we consider a definition.

A ring $(R, +, \times)$ is a set endowed with two operations such that
  1. $(R, +)$ is a commutative group.
  2. $(R,\times)$ is associative i.e. $a(bc)=(ab)c=abc$, $\forall a,b,c \in R$.
  3. $+$ and $\times$ distribute over each other i.e. $a(b+c) = ab + ac$ and $(a+b)c = ac + bc$, for all $a,b,c \in R$.

Note that we implicitly assume that the ring is commutative ( $ab = ba, \forall a,b \in R$ ) and has a multiplicative identity. But with that, we know exactly what a (commutative) ring is !