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Unit 14: Rings
5. If m and n are arbitrary integers and a, b R then (n + m) a = na + ma and n(a + b) = Notes
(a) na + nb (b) a + b n
n
(c) nab + nba (d) an + bn -1
14.4 Summary
In this unit we discussed the following points.
Definition and examples of a ring.
Some properties of a ring like
a . 0 = 0 = 0 . a,
a( b) = (ab) = ( a) b,
( a) ( b) = ab,
a(b c) = ab ac,
(b c)a = ba ca
a, b, c in a ring R.
The laws of indices for addition and multiplication, and the generalised distributive law.
Commutative rings, rings with unity and commutative rings with unity.
Henceforth, we will always assume that a ring means a commutative ring, unless otherwise
mentioned.
14.5 Keywords
Ring: A non-empty set R together with two binary operations, usually called addition (denoted
by f) and multiplication (denoted by .), is called a ring if the following axioms are satisfied.
Commutative Rings: We say that a ring (R, +, .) is commutative if . is commutative, i.e., if ab = ba
for all a, b R. For example, Z, Q and R are commutative rings.
14.6 Review Questions
*
1. Write out the Cayley tables for addition and multiplication in Z , the set of non-zero
6
*
elements of Z . Is (Z , ,'.) a ring? Why?
6
6
2. Show that the set Q 2Q {p 2q|p,q Q} is a ring with respect to addition and
multiplication of real numbers.
a 0
3. Let R = 0 b a,b are real numbers . Show that R is a ring under matrix addition and
multiplication.
a 0
4. Let R = a,b are real numbers . Prove that R is a ring under matrix addition and
b 0
multiplication.
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