Rings and Fields

Rings in Discrete Mathematics

The ring is a type of algebraic structure (R, +, .) or (R, *, .) which is used to contain non-empty set R. Sometimes, we represent R as a ring. It usually contains two binary operations that are multiplication and addition.

An algebraic system is used to contain a non-empty set R, operation o, and operators (+ or *) on R such that:

• (R, 0) will be a semigroup, and (R, *) will be an algebraic group.
• The operation o will be said a ring if it is distributive over operator *.

We have some postulates that need to be satisfied. These postulates are described as follows:

R1

The algebraic group is described by the system (R, +). So it contains some properties, which is described as follows:

1. Closure Property

In the closure property, the set R will be called for composition '+' like this:

x ∈R, y ∈R => x+y ∈ R for all x, y ∈ R

2. Association

In association law, the set R will be related to composition '+' like this:

(x+y) + z = x + (y+z) for all x, y, z ∈ R.

3. Existence of identity

Here, R is used to contain an additive identity element. That element is known as zero elements, and it is denoted by 0. The syntax to represent this is described as follows:

x+ y = x = 0 + x, x ∈ R

4. Existence of inverse

In existence of inverse, the elements x ∈ R is exist for each x ∈ R like this:

x + (-x) = 0 = (-x) + x

5. Commutative of addition

In the commutative law, the set R will represent for composition + like this:

x + y = y + x for all x, y ∈ R

R2

Here, the set R is closed under multiplication composition like this:

xy ∈ R

R3

Here, there is an association of multiplication composition like this:

(x.y).z = x.(y.z) for all x, y, z ∈ R

R4

There is left and right distribution of multiplication composition with respect to addition, like this:

Right distributive law

(y + z). x = y.x + z.x

Left distributive law

x.(y + z) = x.y + x.z

Types of Ring

There are various types of rings, which is described as follows:

Null ring

A ring will be called a zero ring or null ring if singleton (0) is using with the binary operator (+ or *). The null ring can be described as follows:

0 + 0 = 0 and 0.0 = 0

Commutative ring

The ring R will be called a commutative ring if multiplication in a ring is also a commutative, which means x is the right divisor of zero as well as the left divisor of zero. The commutative ring can be described as follows:

x.y = y.x for all x, y ∈ R

The ring will be called non-commutative ring if multiplication in a ring is not commutative.

Ring with unity

The ring will be called the ring of unity if a ring has an element e like this:

e.x = x.e = x for all R

Where

e can be defined as the identity of R, unity, or units elements.

Ring with zero divisor

If a ring contains two non-zero elements x, y ∈ R, then the ring will be known as the divisor of zero. The ring with zero divisors can be described as follows:

y.x = 0 or x.y = 0

Where

x and y can be said as the proper divisor of zero because in the first case, x is the right divisor of zero, and in the second case, x is the left divisor of zero.

0 is described as additive identity in R

Ring without zero divisor

If products of no two non-zero elements is zero in a ring, the ring will be called a ring without zero divisors. The ring without zero elements can be described as follows:

xy = 0 => x = 0 or y = 0

Properties of Rings

All x, y, z ∈ R if R is a ring

1. (-x)(-y) = xy
2. x0 = 0x = 0
3. (y-z)x = yx- zx
4. x(-y) = -(xy) = (-x)y
5. x(y-z) = xy - xz

Field – A non-trivial ring R wit unity is a field if it is commutative and each non-zero element of R is a unit . Therefore a non-empty set F forms a field .r.t two binary operations + and . if 
 

1. For all a, b F, a+bF, 
2. For all a, b, c F a+(b+c)=(a+b)+c, 
3. There exists an element in F, denoted by 0 such that a+0=a for all a F 
4. For every a R there exists an y R such that a+y=0. y is usually denoted by (-a) 
5. a+b=b+a for all a, b F. 
6. a.b F for all a.b F. 
7. a.(b.c)=(a.b).c for all a, b F 
8. There exists an element I in F, called the identity element such that a.I=a for all a in F 
9. For each non-zero element a in F there exists an element, denoted by in F such that =I. 
10. a.b =b.a for all a, b in F . 
11. a.(b+c) =a.b + a.c for all a, b, c in F 
     

Examples – The rings (, +, .), (, + . .) are familiar examples of fields. 

Some important results: 

1. A field is an integral domain. 
2. A finite integral domain is a field. 
3. A non trivial finite commutative ring containing no divisor of zero is an integral domain