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:

  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 R
  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 R
  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, +, .).

R is said to be a commutative ring if the multiplication is commutative.

Some Examples –



  1. (\mathbb{Z}, + ) is a commutative group .(\mathbb{Z}, .) is a semigroup. The disrtributive law also holds. So, ((\mathbb{Z}, +, .) is a ring.
  2. Ring of Integers modulo n: For a n\epsilon \mathbb{N} 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 distriutive laws hold. So ((\mathbb Z_n, +, .) 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 .

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 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 \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 F
  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 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 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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