Page 166 - DMTH403_ABSTRACT_ALGEBRA

Unit 16: Ideals

But, it is valid for R/I if we add some conditions on I: What should these conditions be? To Notes
answer this, assume that the multiplication in (2) is well defined.
Then, (r + I). (0 + I) = r . 0 + I = 0 + I = I for all r  R.
Now, you know that if x  I, then x + I = 0 + I = I.
As we have assumed that is well defined, we get

(r + I) (x + I) = (r + I) . (0 + I) = 0 + I whenever r  R, x  I.
i.e., rx + I = I whenever r  R, x  I
Thus, rx  I, whenever r  R, x  I.
So, if ‘ . ’ is well defined we see that the subring I must satisfy the additional condition that
rx  I whenever r  R and x  I.
We will prove that this extra condition on I is enough to make the operation a well defined one
and (R/I, +, .) a ring. In this unit we will consider the subrings I of R on which we impose the
condition rx  I whenever r  R and x  I.
Definition: 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.
Over here we would like to remark that we are always assuming that our rings are commutative.
In the case of non-commutative rings the definition of an ideal is partially modified as follows.

A non-empty subset I of a non-commutative ring R is an ideal if
(i) a – b  I  a, b  I, and
(ii) ra  I and ar  I  a  I, r  R.
Now let us go back to commutative rings. From the definition we see that a subring I of a ring
R is an ideal of R iff ra  I  r  R a and a  I.
Let us consider some examples. You saw that for any ring R, the set (0) is a subring. In fact, it is
an ideal of R called the trivial ideal of R. Other ideals, if they exist, are known as non-trivial
ideals of R.
You can also verify that every ring is an ideal of itself. If an ideal I of a ring R is such that I  R,
then I is called a proper ideal of R.
For example, if n  0,1, then the subring nZ = { nm | m  Z ) is a proper non-trivial ideal of Z. This
is because for any z  Z and nm  nZ, z(nm) = n(zm)  nZ.

Example: Let X be an infinite set. Consider I, the class of all finite subsets of X. Show that
I is an ideal of  (X).
Solution: I = { A | A is a finite subset of X }. Note that
(i)   I, i.e., the zero element of (X) is in I,
(ii) A – B = A + (–B) = A + B, as B = –B in (X) = A  B.
Thus, if A, B  I, then A - B is again a finite subset of X, and hence A – B  I.

(iii) AB = A  B. Now, whenever A is a finite subset of X and B is any element of  (X), AB
is a finite subset of X. Thus, A  I and B  P (X)  AB  I .
Hence, I is an ideal of (X).

LOVELY PROFESSIONAL UNIVERSITY 159