Page 29 - DMTH403_ABSTRACT_ALGEBRA

Abstract Algebra

Notes As you are familiar with the equation such as a(b + c) = ab + ac and (b + c)a = bc +ba  a, b, c  R.

As this equation explains that multiplication distributes over addition in R. In general we can
define this as.
Definition: If o and * are two binary operations on a set S, then we say that * is distributive over
o if  a, b, c  S, we have a * (b o c) = (a * b) o (a * c) and (b o c) * a = (b * a) o (c * a).

a  b  b c  ab ac


For example, let a * b =  a, b  R. Then a(b a c) = a    = ab * ac, and (b * c)a
2  2  2


=   b c   a  ba ca  ba * ca  a,b,c R.

 2  2
Hence, multiplication is distributive over *.
Let us now look deeper at some binary operations. You know that, for any a " R, a + 0 =a, 0 + a
= a and a + (–a) = (–a) + a = 0. We say that 0 is the identity element for addition and (–a) is the
negative or additive inverse of a.
Definition: Let *.be a binary operation on a set S. If there is an element e  S such that  a  S,
a * e = a and e * a = a, then e is called an identity element for *.
For a  S, we say that b  S is an inverse of a, if a * b = e and b * a = e. In this case we usually write
b = a .
–1
Let us first discuss the uniqueness of identity element for *, and uniqueness of the inverse of an
element with respect to *, if it exists. After that we will discuss the examples related to identity
elements.
Theorem 1: Let * be a binary operation on a set S. Then
(a) if * has an identity element, it must be unique.
(b) if * is associative and s  S has an inverse with respect to *, it must be unique.

Proof: (a) Suppose e and e’ are both identity elements for *.
Then e = e * e’, since e’ is an identity element.
= e’, since e is an identity element.
That is, e = e’. Hence, the identity element is unique.
(b) Suppose there exist a, b  S such that s * a = e = a * s and s * b = e = b * s, e being the identity
element for *, Then
a = a * e = a * ( s * b )
= (a * s) * b, since * is associative.

= e * b = b .
That is, a = b.
Hence, the inverse of s is unique.

This uniqueness theorem allows us to say the identity element and the inverse, henceforth.
A binary operation may or may not have an identity element. For example, the operation of
addition on N has no identity element.

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