Page 170 - DMTH403_ABSTRACT_ALGEBRA
P. 170

Unit 16: Ideals





                                    Figure 16.1:  The  Ideal  Hierarchy                         Notes

































          16.1 Quotient Rings


          You have studied quotient groups. You know that given a normal subgroup N of a group G, the
          set of all cosets of N is a group and is called the quotient group associated with the normal
          subgroup N. Using ideals, we will now define a similar concept for rings. At the beginning we
          said that if (R, +, .) is a ring and I is a subring of R such that
          (R/I, +, .) is a ring, where + and . are defined by

          (X + I) + (y + I) = (x + y) + I and
          ( x + I ) . ( y + I ) = x y + I    x + I , y + I  R / I ,

          then the subring I should satisfy the extra condition that rx  I whenever r  R and x  I, i.e.,
          I should be an ideal. We now show that if I satisfies this extra condition then the operations that
          we have defined on R/I are well defined.

          From group theory we know that (R/I, +) is an abelian group. So we only need to check that is
          well defined, i.e., if
          a + I = a’ + I, b + I = b’ + I, then ab + I = a’b’ + I.

          Now, since a + I = a’ + I, a – a’  I.
          Let a – a’ = x. Similarly, b – b’  1, say b – b’ = y.
          Then ab = (a’ + x ) (b’ + y ) = a’b’ + (xb’+ a’y + xy).

           ab – a’b’  I, since x  I. y  I and I is an ideal of R.
           ab + I = a’b’ + I.






                                           LOVELY PROFESSIONAL UNIVERSITY                                  163
   165   166   167   168   169   170   171   172   173   174   175