# Mathematics | Rings, Integral domains and Fields

Prerequisite – Mathematics | Algebraic Structure

**Ring –** Let addition (+) and Multiplication (.) be two binary operations defined on a non empty set R. Then R is said to form a ring w.r.t addition (+) and multiplication (.) if the following conditions are satisfied:

- (R, +) is an abelian group ( i.e commutative group)
- (R, .) is a semigroup
- For any three elements a, b, c R the left distributive law a.(b+c) =a.b + a.c and the right distributive property (b + c).a =b.a + c.a holds.

Therefore a non- empty set R is a ring w.r.t to binary operations + and . if the following conditions are satisfied.

- For all a, b R, a+bR,
- For all a, b, c R a+(b+c)=(a+b)+c,
- There exists an element in R, denoted by 0 such that a+0=a for all a R
- For every a R there exists an y R such that a+y=0. y is usually denoted by -a
- a+b=b+a for all a, b R.
- a.b R for all a.b R.
- a.(b.c)=(a.b).c for all a, b R
- For any three elements a, b, c R a.(b+c) =a.b + a.c and (b + c).a =b.a + c.a. And the ring is denoted by (R, +, .).

**Some Examples –**

- (, + ) is a commutative group .(, .) is a semi-group. The distributive law also holds. So, ((, +, .) is a ring.

**Ring of Integers modulo n:**For a n[Tex]\mathbb{N} [/Tex]let be the classes of residues of integers modulo n. i.e ={).

(, +) is a commutative group ere + is addition(mod n).

(, .) is a semi group here . denotes multiplication (mod n).

Also the distributive laws hold. So ((, +, .) is a ring.- The set S = {0, 1, 2, 3, 4} is a ring with respect to operation addition modulo 5 & multiplication modulo 5.

(S,+5) is an Abelian Group. From the above 1st composition table we can conclude that (S,+5) satisfies –

- Closure : a ∈ S ,b ∈ S => a +
_{5}b ∈ S ; ∀ a,b ∈ S - Associativity : (a+
_{5}b)+5c = a+_{5}(b+_{5}c) ; ∀ a,b,c ∈ S. - Existence of identity 0 : (a+
_{5}b)+_{5}c = a+_{5}(b+_{5}c) ; ∀ a,b,c ∈ S. - Existence of inverse: Inverse of 0, 1, 2, 3, 4 are 0, 4, 3, 2 , 1 respectively &
- Commutative : (a+
_{5}b) = (b+_{5}a) ; ∀ a,b ∈ S

2. (S,*_{5}) is an Semi Group. From the above 2nd composition table we can conclude that (S,*_{5}) satisfies :

- Closure : a ∈ S ,b ∈ S => a *
_{5}b ∈ S ; ∀ a,b ∈ S - Associativity : (a*
_{5}b)*_{5}c = a*_{5}(b*_{5}c) ; ∀ a,b,c ∈ S

3. Multiplication is distributive over addition :

(a) Left Distributive : ∀ a, b, c ∈ S :

a*_{5} (b +_{5} c)

= [ a * (b + c) ] mod 5

= [a*b + a*c] mod 5

= (a *_{5} b) +5 (a *_{5} c)

⇒ Multiplication modulo 5 is distributive over addition modulo 5.

Similarly , Right Distributive law can also be proved.

So, we can conclude that (S,+,*) is a Ring.

Many other examples also can be given on rings like (, +, .), (, +, .) and so on.

Before discussing further on rings, we define **Divisor of Zero in A ring**and the concept of **unit**.

**Divisor of Zero in A ring –**

In a ring R a non-zero element is said to be divisor of zero if there exists a non-zero element b in R such that a.b=0 or a non-zero element c in R such that c.a=0 In the first case a is said to be a left divisor of zero and in the later case a is said to be a right divisor of zero . Obviously if R is a commutative ring then if a is a left divisor of zero then a is a right divisor of zero also .

**Example –** In the ring (, +, .) are divisors of zero since

and so on .

On the other hand the rings (, +, .), (, +, .), (, +, .) contains no divisor of zero .

**Units –**

In a non trivial ring R( Ring that contains at least to elements) with unity an element a in R is said to be an unit if there exists an element b in R such that a.b=b.a=I, I being the unity in R. b is said to be multiplicative inverse of a.

Some Important results related to Ring:

- If R is a non-trivial ring(ring containing at least two elements ) withunity I then I 0.
- If I be a multiplicative identity in a ring R then I is unique .
- If a be a unit in a ring R then its multiplicative inverse is unique .
- In a non trivial ring R the zero element has no multiplicative inverse .

**Types of Ring :**

**Null Ring**: The singleton set : {0} with 2 binary operations ‘+’ & ‘*” defined by :

0+0 = 0 & 0*0 = 0 is called zero/ null ring.**Ring with Unity**: If there exists an element in R denoted by 1 such that :

1*a = a* 1 = a ; ∀ a ∈ R, then the ring is called Ring with Unity.**Commutative Ring**: If the multiplication in the ring R is also commutative, then ring is called a commutative ring.**Ring of Integers**: The set I of integers with 2 binary operations ‘+’ & ‘*’ is known as ring of Integers.**Boolean Ring**: A ring whose every element is idempotent, i.e. , a^{2}= a ; ∀ a ∈ R

Now we introduce a new concept Integral Domain.

**Integral Domain –** A non -trivial ring(ring containing at least two elements) with unity is said to be an integral domain if it is commutative and contains no divisor of zero ..

**Examples –**

The rings (, +, .), (, +, .), (, +, .) are integral domains.

The ring (2, +, .) is a commutative ring but it neither contains unity nor divisors of zero. So it is not an integral domain.

Next we will go to Field .

**Field –** A non-trivial ring R with 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

- For all a, b F, a+bF,
- For all a, b, c F a+(b+c)=(a+b)+c,
- There exists an element in F, denoted by 0 such that a+0=a for all a F
- For every a R there exists an y R such that a+y=0. y is usually denoted by (-a)
- a+b=b+a for all a, b F.
- a.b F for all a.b F.
- a.(b.c)=(a.b).c for all a, b F
- There exists an element I in F, called the identity element such that a.I=a for all a in F
- For each non-zero element a in F there exists an element, denoted by in F such that =I.
- a.b =b.a for all a, b in F .
- 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:

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