IDA: Example

Examples

Here are seven examples, most of which you have already seen.

Usual arithmetic

Modular arithmetic

Polynomial rings

Residue class rings

Matrix rings

The Gaussian integers

The Quaternions

Z, Q, R, C with the usual addition and multiplication are all commutative rings.

Addition and multiplication as defined in Chapter 2 define a commutative ring structure on Z/nZ. The zero element is the class of 0, the unit element is the class of 1.

Z[X], Q[X], R[X], C[X] with the usual addition and multiplication are commutative rings.

If R = Z, Q, R or C, and f is a polynomial in R[X], then R[X]/(f) as defined in Chapter 4 is a commutative ring. The zero element is 0 + (f), the unit element is 1 + (f).

The subset R = Z + Zi of the complex numbers is a ring with the usual addition and multiplication, with zero element 0 (= 0 + 0i) and unit element 1 (= 1 + 0i). Most ring properties, like associativity, are `inherited' from the ring C: since they hold in the complex numbers they hold a fortiori in the subset R.
Gauss

A point that is easily overlooked is that we have to check that R is closed with respect to the operations. For instance,

(a + bi) * (c + di) = (ac - bd) + (ad + bc)i

shows that the set R is closed with respect to multiplication, since ac - bd and ad + bc are integers if a, b, c, d are. The ring R is called the ring of Gaussian integers.

The structure

(M_n(R),+,*,0,1),

where M_n(R) is the set of n by n matrices with coefficients in R, 0 is short for the zero matrix, 1 is short for the identity matrix, + denotes matrix addition and * denotes matrix multiplication.

If n > 1, it is easy, and left to the reader, to find matrices A, B such that A * B and B * A are distinct. Thus, M_n(R) is not commutative for n > 1.

Let R be Z, Q or R. Take e, i, j, k to be a set of four vectors (think of a standard basis), and consider the set

H = R + Ri + Rj + Rk.

On H we define the operations + and * as follows. Let x = ae + bi + cj + dk and x' = a'e + b'i + c'j + d'k, then

x + x' = (a + a')e + (b + b')i + (c + c')j + (d+d')k,

x * x' = (aa' - bb' - cc' - dd')e + (ab' + ba' + cd' - dc')i + (ac' - bd' + ca' + db')j + (ad' + bc' - cb' + da')k.

Now H is a ring. (It is quite tedious to check associativity, etc.) Since ij = k and ji = -k, the ring is not commutative.
H is called the ring of (integral, rational or real, respectively) quaternions. The letter H is used here as this ring was first studied by the mathematician Hamilton.