Page 173 - DMTH403_ABSTRACT_ALGEBRA

Abstract Algebra

Notes 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.
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 nrn  nZ, z(nm) = n(zm)  nZ.
An element a of a ring R is called nilpotent if there exists a positive integer n such that

a” = 0.

2 2
0
9
For example, 3 and 6 are nilpotent elements of Z , since 3   and 6  36  0. Also,
 9
in any ring R, 0 is a nilpotent element.
16.3 Keywords
Proper Ideal: 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
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.
Nilpotent: An element a of a ring R is called nilpotent if there exists a positive integer n such that
a” = 0.

Quotient Group: 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.
This ring is called the quotient ring of R by the ideal I.

16.4 Review Questions

1. Let S be a subring of a ring R. Can we always define a ring homomorphism whose domain
is R and kernel is S? Why?
2. Prove Theorem 8.

3. In the situation of Theorem 8 prove that
(a) if g o f is 1 – 1, then so is f.
(b) if g o f is onto, then so is g.

4. Use Theorem 8 to show that the function h : Z × Z  Z defined by h((n, m)) = m is a
2
homomorphism.
5. Which of the following functions are ring isomorphisms?
(a) f : Z  : f(n) = n

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