DHARM92 MATHEMATICAL FOUNDATION OF COMPUTER SCIENCEReader must note that if we change the first group as (R, +), then R and R+ are isomor-
phism to each other. Because, clearly mapping f : R → R+ is surjective. Let x, y ∈ R, so f(x + y)
= ex+y = ex ey, and f(x) × f(y) = ex × ey = ex+y. So, f(x + y) = f(x) × f(y). Therefore, groups (R, +) and
(R+, ×) are isomorphism to each other.4.12 Rings...................................................................................................................................
In the previous sections we have discussed the algebraic systems that are defined with one
binary operation only. In this section we will discuss different algebraic systems viz. rings,
fields, etc. that are defined with the composition of two binary operations additions and mul-
tiplication denoted respectively by + and •.Ring
Let X be an nonempty set and + and • are two binary addition and multiplication operations
on X, then algebraic system (X, +, •) is called a ring, if it holds following properties :
- (X, +) is a Abelian group,
[Closure, Associative, identity and Inverse law for addition and Commuta-
tive] - (X, •) is a semigroup, and
[Closure and Associative law for multiplication] - The operation • is distributive over the operation +, i.e., for any x, y and z ∈ X
x • (y + z) = (x • y) + (x • z) [Left Distributive law]
or, (y + z) • x = (y • x) + (z • x) [Right Distributive law]
Property 1 assert that algebraic structure (X, +) is a abelian or commutative group,
whose natural element will be denoted by 0, i.e.,
0 = x + (– x) for all x ∈ X, and x + 0 = x for all x ∈ X.
So, (X, +) is marked as the additive group of the ring.
In addition to the properties 1, 2, and 3 if commutative law holds for multiplication
also, i.e.,
x • y = y • x for all x, y ∈ X
then (X, +, •) is called a commutative ring.
Further if (X, •) has an identity element 1. It means, once semigroup becomes monoid,
then ring (X, +, •) is called a ring with unity.
Assume X is the set of real numbers (R) or set of integers (I) then, algebraic system
(R, +, •) and (I, +, •) are the examples of rings. Since, (I, +) is a Abelian group, (I, •) is a
semigroup and operation • is distributive over operation + in the set I. Hence, (I, +, •) is a
ring. Further, algebraic structure (I, •) has an identity element 1 so (I, •) is also a monoid.
Therefore, (I, +, •) is a ring with unity. It is also a commutative ring, because operation
multiplication (•) holds commutativity over I.
Example 4.14. Let set X = 2I where I is the set of integers, i.e., X = { ....-4, -2, 0, 2, 4, .....}, then
(X, +, .) is not a ring with unity, but it is a commutative ring.
Sol. Since X is the set of even integers. We first check whether algebraic system (I, +, •) is a
ring. Since, (X, +) is a Abelian group, (X, •) is a semigroup, and operation • is distributive over
- hence, (X, +, •) is a ring.