# 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 :

- There exists an element denoted by such that . 0 is called the
**additive identity**or the**zero element**in R. - For each , there exists an element such that . b is called
**additive inverse**or**negative**of a and is written as b=-a so that a+(-a)=0. - (
**Left distributive law.**) - (
**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 ring is called **boolean** if . A ring R is called a ring with a **unit element** or **unity** or **identity** if an element such that . Let R be a ring with unit element e. An element is called **invertible**, if there exists an element such that . If n is a positive integer and a an element of a ring R then we define and .

## Examples

editOne of the most important rings is the ring of integers 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 is an example of a ring without identity. Like , the sets of rational numbers , of real numbers and of complex numbers are also rings with identity. However is not a ring.

The **ring of Gaussian integers** is given by the set 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

editWe 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:

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

For example,

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,

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*:

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

If and , then:

Like any congruence relation, congruence modulo *n* is an equivalence relation, and the equivalence class of the integer *a*, denoted by , is the set . 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 .

The set of congruence classes modulo *n* is denoted as (or, alternatively, or ) and defined by:

When *n* ≠ 0, has *n* elements, and can be written as:

We can define addition, subtraction, and multiplication on by the following rules:

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

In this way, becomes a commutative ring. For example, in the ring , we have

as in the arithmetic for the 24-hour clock.