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Mathematics | Rings, Integral domains and Fields

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  • Last Updated : 19 May, 2022
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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: 

  1. (R, +) is an abelian group ( i.e commutative group) 
  2. (R, .) is a semigroup 
  3. For any three elements a, b, c \epsilon   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 \epsilon   R, a+b\epsilon   R, 
  2. For all a, b, c \epsilon   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 \epsilon
  4. For every a \epsilon   R there exists an y \epsilon   R such that a+y=0. y is usually denoted by -a 
  5. a+b=b+a for all a, b \epsilon   R. 
  6. a.b \epsilon   R for all a.b \epsilon   R. 
  7. a.(b.c)=(a.b).c for all a, b \epsilon
  8. For any three elements a, b, c \epsilon   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 – 
 

  1. (\mathbb{Z}   , + ) is a commutative group .(\mathbb{Z}   , .) is a semi-group. The distributive law also holds. So, ((\mathbb{Z}   , +, .) is a ring. 
     
  2. Ring of Integers modulo n: For a n\epsilon   [Tex]\mathbb{N}   [/Tex]let \mathbb Z_n   be the classes of residues of integers modulo n. i.e \mathbb Z_n   ={\bar{0}, \bar{1}, \bar{2}, ......., \overline{n-1}   ). 
    (\mathbb Z_n   , +) is a commutative group ere + is addition(mod n). 
    (\mathbb Z_n   , .) is a semi group here . denotes multiplication (mod n). 
    Also the distributive laws hold. So ((\mathbb Z_n   , +, .) 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 (\mathbb R   , +, .), (\mathbb Q   , +, .) 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 (\mathbb Z_6   , +, .) \bar{2}, \bar{3}, \bar{4}   are divisors of zero since 
\bar{2}.\bar{3}=\bar{6}=\bar{0}   and so on . 
On the other hand the rings (\mathbb Z   , +, .), (\mathbb R   , +, .), (\mathbb Q   , +, .) 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 \neq   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 (\mathbb Z   , +, .), (\mathbb R   , +, .), (\mathbb Q   , +, .) are integral domains. 
The ring (2\mathbb Z   , +, .) 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 \epsilon   F, a+b\epsilon   F, 
  2. For all a, b, c \epsilon   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 \epsilon
  4. For every a \epsilon   R there exists an y \epsilon   R such that a+y=0. y is usually denoted by (-a) 
  5. a+b=b+a for all a, b \epsilon   F. 
  6. a.b \epsilon   F for all a.b \epsilon   F. 
  7. a.(b.c)=(a.b).c for all a, b \epsilon
  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 a^{-1}   in F such that a a^{-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 (\mathbb Q   , +, .), (\mathbb R   , + . .) 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 

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