Page 172 - DMTH403_ABSTRACT_ALGEBRA

Unit 16: Ideals

Solution: I = 4 R, and hence is an ideal of R. From group theory you know that the number 8 of Notes
o(R) 8
elements in R/I = o(R/I) =   4.
o(I) 2
You can see that these elements are

0 + 1 = {0, 4}, 1 + 1 = {1, 5} , 2 + 1 = {2, 6}, 3 + I = (3, 7}.
The Cayley tables for + and . in R/I are

Self Assessment

1. A non-empty subset I of a ring (R+,.) an ................. of R of a – b  I for all a, b  I.
(a) ring (b) subring

(c) polynomial (d) ideal
2. If n  0, 1. Then the subring nZ = {nm | m  Z} is a proper ................. ideal of Z.
(a) non-trivial (b) trivial
(c) direct (d) indirect

3. X be a set and Y be a non-empty subset of X. Then I = {A  (x) | A ................. y = } is an ideal
of (x).
(a)  (b) 

(c)  (d) 
4. If I and J are ideals of a ring R, then I ................. J are ideals ring R.
(a)  (b) 
(c)  (d) 

5. A normal subgroup N of a group G, the set of all cosets of N is a group and is called
................. associated with the normal subgroup N.
(a) quotient group (b) ring

(c) subring (d) ideal
16.2 Summary

We call a non-empty subset I of a ring (R, +, .) an ideal of R if

(i) a – b  I for all a, b  I, and
(ii) ra  I for all r  R and a  I.

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