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 &minus; 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.