Ring Theory/Rings

We will start by the definition of a ring.

Definition 1: A ring is a non empty set R together with two binary compositions defined by + and ., and satisfying the following properties hold for any $a,b,c\in R$:

• $a+b\in R$
• a + b = b + a
• a + (b + c) = (a + b) + c
• There exists an element denoted by $0\in R$ such that a + 0 = a. 0 is called the additive identity or the zero element in R.
• For each $a\in R$, there exists an element $b\in R$ such that a + b = 0. b is called additive inverse or negative of a and is written as b=-a so that a+(-a)=0.
• $a.b\in R$
• a.(b.c) = (a.b).c
• a.(b + c) = a.b + a.c (Left distributive law.)
• (a + b).c = a.c + b.c (Right distributive law.)

We denote a ring by (R,+,.). When the context is clear we just talk about a ring R and assume that the operations + and . are implicit. We will also drop the . in the operation a.b and just say ab.

The first 5 axioms of a ring just mean that (R,+) is an abelian group. The next two mean that (R,.) is a semi group. A ring is called commutative if $a.b=b.a\ \forall a,b\in R$. A ring is called boolean if $x^2=x\ \forall x\in R$. A ring R is called a ring with a unit element or unity or identity if $\exists$ an element $e\in R$ such that $ae=ea=a\ \forall a\in R$. Let R be a ring with unit element e. An element $a\in R$ is called invertible, if there exists an element $b\in R$ such that ab = ba = e. If n is a positive integer and a an element of a ring R then we define $a^n=\underbrace{aa\cdots a}_{n\ times}$ and $na=\underbrace{a+a\cdots +a}_{n\ times}$.

Examples

One of the most important rings is the ring of integers $\mathbb{Z}$ with usual addition and multiplication playing the roles of + and . respectively. It is a commutative ring with identity as 1. The set of even numbers $2\mathbb{Z}:=\{0,\pm 2,\pm 4\cdots\}$ is an example of a ring without identity. Like $\mathbb{Z}$, the sets of rational numbers $\mathbb{Q}$, of real numbers $\mathbb{R}$ and of complex numbers $\mathbb{C}$ are also rings with identity. However $\mathbb{N}$ is not a ring.

The ring of Gaussian integers is given by the set $\mathbb{Z}[i]=\{m+ni:m,n\in\mathbb{Z}\}$ where usual addition and multiplication of complex numbers are the operations. Here i stands (0,1) as is usual in the complex plane.

The set of all n by n matrices with real entries is an example of a non commutative ring with identity, under the usual addition and multiplication of matrices.

The ring of integers modulo n

We now digress slightly to discuss a special kind of an equivalence relation which gives rise to an important class of finite rings.

Let n be a positive integer. Two integers a and b are said to be congruent modulo n, if their difference a − b is an integer multiple of n. If this is the case, it is expressed as:

$a \equiv b \pmod n.\,$

The above mathematical statement is read: "a is congruent to b modulo n".

For example,

$38 \equiv 14 \pmod {12}\,$

because 38 − 14 = 24, which is a multiple of 12. For positive n and non-negative a and b, congruence of a and b can also be thought of as asserting that these two numbers have the same remainder after dividing by the modulus n. So,

$38 \equiv 2 \pmod {12}\,$

because both numbers, when divided by 12, have the same remainder (2). Equivalently, the fractional parts of doing a full division of each of the numbers by 12 are the same: .1666... (38/12 = 3.166..., 2/12 = .1666...). From the prior definition we also see that their difference, a - b = 36, is a whole number (integer) multiple of 12 ( n = 12, 36/12 = 3).

The same rule holds for negative values of a:

$-3 \equiv 2 \pmod 5.\,$

The properties that make this relation a congruence relation (respecting addition, subtraction, and multiplication) are the following.

If $a_1 \equiv b_1 \pmod n$ and $a_2 \equiv b_2 \pmod n$, then:

• $(a_1 + a_2) \equiv (b_1 + b_2) \pmod n\,$
• $(a_1 - a_2) \equiv (b_1 - b_2) \pmod n\,$
• $(a_1 a_2) \equiv (b_1 b_2) \pmod n.\,$

Like any congruence relation, congruence modulo n is an equivalence relation, and the equivalence class of the integer a, denoted by $\overline{a}_n$, is the set $\left\{\ldots, a - 2n, a - n, a, a + n, a + 2n, \ldots \right\}$. This set, consisting of the integers congruent to a modulo n, is called the congruence class or residue class of a modulo n. Another notation for this congruence class, which requires that in the context the modulus is known, is $\displaystyle [a]$.

The set of congruence classes modulo n is denoted as $\mathbb{Z}/n\mathbb{Z}$ (or, alternatively, $\mathbb{Z}/n$ or $\mathbb{Z}_n$) and defined by:

$\mathbb{Z}/n\mathbb{Z} = \left\{ \overline{a}_n | a \in \mathbb{Z}\right\}.$

When n ≠ 0, $\mathbb{Z}/n\mathbb{Z}$ has n elements, and can be written as:

$\mathbb{Z}/n\mathbb{Z} = \left\{ \overline{0}_n, \overline{1}_n, \overline{2}_n,\ldots, \overline{n-1}_n \right\}.$

We can define addition, subtraction, and multiplication on $\mathbb{Z}/n\mathbb{Z}$ by the following rules:

• $\overline{a}_n + \overline{b}_n = \overline{a + b}_n$
• $\overline{a}_n - \overline{b}_n = \overline{a - b}_n$
• $\overline{a}_n \overline{b}_n = \overline{ab}_n.$

The verification that this is a proper definition uses the properties given before.

In this way, $\mathbb{Z}/n\mathbb{Z}$ becomes a commutative ring. For example, in the ring $\mathbb{Z}/24\mathbb{Z}$, we have

$\overline{12}_{24} + \overline{21}_{24} = \overline{9}_{24}$

as in the arithmetic for the 24-hour clock.