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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 = ab + 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 ) = ab + (xb+ ay + xy).
ab ab I, since x I. y I and I is an ideal of R.
ab + I = ab + I.
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