L#14 | 2022 | RING THEORY AND LINEAR ALGEBRA-I | RING HOMOMORPHISM | B.Sc. Mathematics

RING HOMOMORPHISM

In our work with groups, we saw that one way to discover information
about a group is to examine its interaction with other groups by way of
homomorphisms. It should not be surprising to learn that this concept
extends to rings with equally profitable results.
Just as a group homomorphism preserves the group operation, a ring
homomorphism preserves the ring operations.
Definitions Ring Homomorphism, Ring Isomorphism
A ring homomorphism f from a ring R to a ring S is a mapping from
R to S that preserves the two ring operations; that is, for all a, b in R,
f(a 1 b) 5 f(a) 1 f(b) and f(ab) 5 f(a)f(b).
A ring homomorphism that is both one-to-one and onto is called a
ring isomorphism.

As is the case for groups, in the preceding definition the operations
on the left of the equal signs are those of R, whereas the operations on
the right of the equal signs are those of S.
Again as with group theory, the roles of isomorphisms and homomor-
phisms are entirely distinct. An isomorphism is used to show that two
rings are algebraically identical; a homomorphism is used to simplify a
ring while retaining certain of its features.
A schematic representation of a ring homomorphism is given in
Figure 15.1. The dashed arrows indicate the results of performing the
ring operations.
The following examples illustrate ring homomorphisms. The reader
should supply the missing details.