Ring Theory

Definition and Examples Of Rings :

 Definition of Rings :

                 A non empty set R is said to be a ring (or an associative ring) if in R there are defined two binary operations called addition and multiplication denoted by  + and * such that for all a,b,c ∈ R .

 1. a+b ∈ R (closure axiom for addition)

 2. a+b = b+a (commutative axiom for                                                                       addition)

 3. a+(b+c) = (a+b)+c   (associative axiom for

                                                       addition)

 4. There exists an element 0 in R such that 

             a+0 = 0+a =a for all a∈ R

       (Existence of additive identity element in R)

5. For every a in R there exists an element         -a  in R such that a + (-a) = (-a) + a = 0

 (Existence of additive inverse element in R)

6. ab ∈ R (closure axiom for multiplication )

7. a(bc) =( ab)c (Associative axiom for                          multiplication )
8.  {a(b+c) = ab + ac
                                           (Distributive Axiom)
      (a+ b)c =ac +bc}

Definition of Commutative Ring :

      A ring R is said to be a Commutative ring if  a.b = b.a for all a,b ∈ R 

Definition of Ring With Unit Element :

        A ring R is said to be a ring with unit element if ∃ an element 1 in R such that 

a.1 = 1.a = a for all a ∈R 

Definition Of Zero Divisor : 

            Let R be a commutative ring . An element a ≠ 0 in R is said to be zero divisor if ∃ b ≠ 0 in R such that ab = 0 .

Definition Of Integral Domain : 

            A commutative ring without zero divisor is called an integral domain .

Definition Of Division Ring :

                 A ring is said to be division ring if its non zero elements form a group under multiplication .

Definition Of Field :

          A field is a commutative division ring.

Characteristic Of Ring :

        Let R be a ring with zero element 0 . The characteristic of R is the least positive integer n such that na = 0 for all a ∈ R . 

         If ∃ number such that positive integer then R is said to be of characteristic zero or infinite .

Homomorphisms : 

          A mapping Φ from a ring R into the ring R' is said to be a homomorphism if 

      (1)   Φ(a+b) = Φ(a) + Φ(b) 

      (2)    Φ(ab) =  Φ(a) . Φ(b) ∀ a , b ∈ R 

Kernel Of Homomorphism : 

       Let Φ be a homomorphism of R into R' . The kernel of Φ is denoted by 

            I(Φ) = { a ∈ R : Φ(a) = 0 , 0 is the zero element of R' }

Ideals : 

           A  non empty subset of U of a ring R is said to be (two sided) ideal of R if 

   (1) U is a subgroup of R under addition .

    (2) ur , ru ∈ U ∀ u ∈ U , r ∈ R 

Maximal Ideal : 

                     An ideal M ≠ R in a ring R is said to be the maximal ideal of R if whenever U is an ideal of R such that M ⊂ U ⊂ R then either R = U or M = U .

Isomorphism : 

                 A homomorphism of R into R' is said to be an isomorphism if it is a one one mapping .

Isomorphic Rings : 

            Two rings are said to be isomorphic if there is a isomorphism of one onto the other .

These are some definitions related to Ring Theory .