Rings, Integral Domain & Field

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: 

1. (R, +) is an abelian group ( i.e commutative group) 
2. (R, .) is a semigroup 
3. For any three elements a, b, c ϵ 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. 

1. For all a, b ϵ R, a+b ϵ R,
2. For all a, b, c ϵ R a+(b+c)=(a+b)+c,
3. There exists an element in R, denoted by 0 such that a+0=a for all a ϵR
4. For every a ϵ R there exists an y ϵ R such that a + y = 0. y is usually denoted by -a
5. a+b=b+a for all a, b ϵ R.
6. a.b ϵ R for all a, b ϵ R.
7. a.(b.c) = (a.b).c for all a, b, c ϵ R
8. For any three elements a, b, c ϵ Ra.(b+c) =a.b + a.c and (b + c).a =b.a + c.a. And the ring is denoted by (R, +, .).  

Some Examples - 

1. (, + ) is a commutative group .(, .) is a semi-group. The distributive law also holds. So, ((, +, .) is a ring.  
2. Ring of Integers modulo n: For a n ϵ 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.
3. 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+₅b)+5c = a+₅(b+₅c) ; ∀ a,b,c ∈ S.
• Existence of identity 0 : (a+₅b)+₅c = a+₅(b+₅c) ; ∀ a,b,c ∈ S.
• Existence of inverse: Inverse of 0, 1, 2, 3, 4 are 0, 4, 3, 2 , 1 respectively &
• Commutative :  (a+₅b) = (b+₅a) ;  ∀ 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*₅b)*₅c = a*₅(b*₅c) ; ∀ 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 ringand 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: 

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

Types of Ring

1. Null Ring : The singleton set : {0} with 2 binary operations '+' & '*" defined by :
   0+0 = 0 & 0*0 = 0 is called zero/ null ring.
2. 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.
3. Commutative Ring : If the multiplication in the ring R is also commutative, then ring is called a commutative ring.
4. Ring of Integers : The set I of integers with 2 binary operations '+' & '*' is known as ring of Integers.
5.  Boolean Ring : A ring whose every element is idempotent, i.e. , a² = 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 

1. For all a, b ϵ F, a + b ϵ F,
2. For all a, b, c ϵ 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 ϵ F
4. For every a ϵR there exists an y ϵ R such that a + y = 0. y is usually denoted by (-a)
5. a + b = b + a for all a, b ϵ F.
6. a.b ϵ F for all a.b ϵ F.
7. a.(b.c)=(a.b).c for all a, b ϵ 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 bya^-1 in F such that aa^-1 = 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: 

• 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