Page 148 - DMTH403_ABSTRACT_ALGEBRA

Unit 14: Rings

14.1 What is a Ring? Notes

You are familiar with Z, the set of integers. You also know that it is a group with respect to
addition. Is it a group with respect to multiplication too? No. But multiplication is associative
and distributes over addition. These properties of addition and multiplication of integers allow
us to say that the system (Z, +, .) is a ring. But, what do we mean by a ring?
Definition: A non-empty set R together with two binary operations, we mean usually called
addition (denoted by f) and multiplication (denoted by .), is called a ring if the following axioms
are satisfied:
R 1) a + b = b + a for all a, b in R, i.e., addition is commutative.

R 2) (a + b) + c = a + (b + c) for all a, b, c in R, is., addition is associative.
R 3) There exists an element (denoted by 0) of R such that
a + 0 = a = 0 + a for all a in R, i.e., R has an additive identity.
R 4) For each a in R, there exists x in R such that a + x =: 0 = x + a, i.e., every elements of R has
an additive inverse.
R 5) (a . b).c = a.(b . c) for all a, b, c in R, i.e., multiplication is associative.
R 6) a.(b + c) = a . b + a . c, and

( a t b ) . i = a . d + b . c
for all a, b, c in R,
i.e., multiplication distributes over addition from the left as well as the right.
The axioms RI-R4 say that (R, +) is an abelian group. The axiom R5 says that multiplication is
associative. Hence, we can say that the system (R, +, .) is a ring if
(i) (R, +) is an abelian group,
(ii) (R, .) is a semigroup, and

(iii) for all a, b, c in R, a.(b + c) = a . b t a . c, and (a + b ) = a . c + b . c.
As you know that the addition identity 0 is unique, and each element a of R has a unique additive
inverse (denoted by - a). We call the element 0 the zero element of the ring.

By convention, we write a – b for a + ( –b).
Let us look at some examples of rings now. You have already seen that Z is a ring. What about
the sets Q and R? Do (Q, +, .) and (R, +, .) satisfy the axioms R1 – R6? They do.

Therefore, these systems are rings.
The following example provides us with another set of examples of rings

Example: Show that (nZ, +, .) is a ring, where n  Z.
Solution: You know that nZ = { nm I m  Z } is an abelian group with respect to addition. You
also know that multiplication in nZ is associative and distributes over addition from the right as
well as the left. Thus, nZ is a ring under the usual addition and multiplication.
So far the examples that we have considered have been infinite rings, that is, their underlying
sets have been infinite sets. Now let us look at a finite ring, that is, a ring (R, +. .) where R is a

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