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Unit 14: Rings




          5.   If m and n are arbitrary integers and a, b  R then (n + m) a = na + ma and n(a + b) =  Notes
               (a)  na + nb                 (b)  a  + b n
                                                  n
               (c)  nab + nba               (d)  an + bn -1
          14.4 Summary


          In this unit we discussed the following points.
               Definition and examples of a ring.
          
               Some properties of a ring like
          
               a . 0 = 0 = 0 . a,
               a(– b) = – (ab) = (– a) b,
               (– a) (– b) = ab,
               a(b – c) = ab – ac,

               (b – c)a = ba – ca
                 a, b, c in a ring R.
               The laws of indices for addition and multiplication, and the generalised distributive law.
          
               Commutative rings, rings with unity and commutative rings with unity.
          
          Henceforth, we will always assume that a ring means a commutative  ring, unless otherwise
          mentioned.

          14.5 Keywords


          Ring: A non-empty set R together with two binary operations, usually called addition (denoted
          by f) and multiplication (denoted by .), is called a ring if the following axioms are satisfied.
          Commutative Rings: We say that a ring (R, +, .) is commutative if . is commutative, i.e., if ab = ba
          for all a, b  R. For example, Z, Q and R are commutative rings.

          14.6 Review Questions


                                                                     *
          1.   Write out the Cayley  tables for addition and multiplication in  Z ,  the set of non-zero
                                                                     6
                               *
               elements of Z . Is  (Z , ,'.)  a ring? Why?
                                 
                               6
                          6
          2.   Show  that  the  set  Q   2Q   {p   2q|p,q  Q} is  a ring  with respect  to addition  and
               multiplication of real  numbers.
                          a 0             
          3.   Let R =   0 b   a,b are real numbers .   Show that R is a ring under matrix addition and
                                          
               multiplication.

                        a    0            
          4.   Let R =      a,b are real numbers .   Prove that R is a ring under matrix addition and
                           b  0           
               multiplication.




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