Ring Algebraic structures
(Monday, May 3, 2010)
Introduction
We have seen that group is an algebraic structures with only one binary operation defined in it. In rings we shall study the algebraic structures equipped with two binary operations. To a certain extent the two operations are mainly addition and multiplication.
Definition: A non-empty set R together with binary operations ‘+’ and ‘.’ defined in it is said to be a ring if the following axioms are satisfied.
R1. Closure property for addition:
" a, b e R, a + b e R
R2. Associative law for addition:
" a,b,ceR, (a+b)+c= a+(b+c)
R3. Existence of identity for addition:
For 0 e R, such that for any a e R,
a + 0 = a = 0 + a
0, being the identity element in R
R4. Existence of inverse of an element:
For any a e R,
a + (-a) = (-a) + a = 0, -a e R
0 being the additive element in R,
(-a) is the additive inverse in R
R5. Commutative law for addition:
For any a, b e R, Þ a + b = b + a
Axioms for multiplication:
R6. Closure property for addition:
" a, b e R, a . b e R
R7. Associative law for addition:
" a, b, c e R, (a . b) . c= a .(b . c)
Axioms for addition and multiplication:
R8. Distributive law:
a . (b + c) = a . b + a . c " a, b e R,
(Left distributive law)
and
(a + b) . c) = a . c + b . c " a, b e R,
(Right distributive law)
Axioms for addition and multiplication:
R8. Distributive law:
a . (b + c) = a . b + a . c " a, b e R,
(Left distributive law)
and
(a + b) . c) = a . c + b . c " a, b e R,
(Right distributive law)
The axioms R1 to R5 merely says states that R is an Abelian or commutative group under the operation addition, i.e., R is additively Abelian group. The axioms R6 and R7 states that R is a semi group under the operation of multiplication, i.e., R is multiplicatively a semi-group. The axiom R8 serves to inter-relate the two operations in R.
Thus, a set R, equipped with two binary operations viz., addition and multiplication is said to form a ring, if
(i) R is additively an Abelian group
(ii) R is multiplicatively a semi-group
(iii) Distributive law holds in R.
The ring is denoted by áR, +, .ñ
The additive identity element 0, called the zero of the ring R is unique and so is the additive inverse -a of an element a. Since, the equation a +x = b has a unique solution in any additive Abelian group, it is observed that subtraction is possible and is unique in a ring R. If the multiplication in a ring R is such that a . b = b . a, for any a, b e R is called a commutative ring.
Example 1:
If R be the set of integers, ‘+’ and ‘.’ is the usual addition and multiplication of in integers, then we can see that R is a ring.
Theorem:
If R has unity element 1, then for any
a e R,
a + (-1) . a = 1 . a + (-1) . a
= [1 + (-1)] . a
= 0 . a
= 0
Whence e get (-1) . a = -a.
In particular if a = -1, we have (-1).(-1)
= - (1) = 1
Note:
The computation with negatives and 0 is the same
a + (-b) as (a-b)
We can now prove that by left distributive law
a. (b - c) = a . b + a .(-c)
= a . b + (-a . c)
= a . b – a . c
and using right distributive law we get
(b – c) . a = b . a – c . a
Example 2:
If R is the set of even integers, then under the usual operation of ‘+’ and ‘.’ R is a commutative ring but has no unity element.
Theorem:
If R is a ring, then for all a , b e R
(i) a . 0 = 0 . a = 0
(ii) a . (-b) = (-a) . b = - (a . b)
(iii) (-a) . (-b) = a . b
If in addition R has unity element 1, then
(iv) (-1) . a = -a
(v) (-1) . (-1) = 1
Definition:
In abstract algebra, a nonzero element a of a ring is a left zero divisor if there exists a nonzero b such that a . b = 0. Right zero divisors are defined analogously, that is, a nonzero element a of a ring is a right zero divisor if there exists a nonzero c such that ca = 0.
An element that is both a left and a right zero divisor is simply called a zero divisor. If multiplication in the ring is commutative, then the left and right zero divisors are the same. A nonzero element of a ring that is neither left nor right zero divisor is called regular.
Example 1:
The ring Z of integers has no zero divisors, but in the ring Z × Z with component wise multiplication, (0,1)·(1,0) = (0,0), so both (0,1) and (1,0) are zero divisors.
Example 2:
An example of a zero divisor in the ring of 2-by-2 matrices is the matrix
Ring with zero divisor:
A ring áR, +, .ñ is said to be a ring with zero divisors, if for some pair of elements, a, b e R, a . b=0 Þ either a¹0 or b¹0, we have a . b = 0.
Ring without zero divisor:
A ring áR, +, .ñ is said to be a ring with zero divisors, if for some pair of elements, a, b e R, a . b=0 Þ either a=0 or b=0, we have a . b = 0, then it is called a ring without divisors of zero.
Integral domain:
A commutative ring having unity element and no zero divisors is called an integral domain.
In other words, an algebraic structure áR, +, .ñ is an integral domain (i) if it’s a commutative ring, (ii) it has unity element and (iii) without a zero divisor.
Example :
The ring of integers is an integral domain. But, the ring of even integers with zero element does not contain the unity element and hence, it is not an integral, although it has divisor of zero.
It is possible to express the condition of a ring R being with or without zero divisors by the help of cancellation laws. We way that the cancellation laws hold in a ring R, if for a, b e R
a¹0, a . b = a . c Þ b = c (left
cancellation law)
a¹0, b . a = c . a Þ b = c (right
cancellation law)
Theorem :
A ring R is without any zero divisor, iff if, the cancellation laws hold in R.
Proof: First, we suppose that R is without any zero divisor. Let a, b, c e, R, such that a. b = a. c, a¹0, then
a . b = a . c Þ a . b - a . c = 0
Þ a. (b – c) = 0
Þ (b – c ) = 0
\b = c
Similarly, we can show that
b . a = c . a Þ (b – c) . a = 0
Þ (b – c ) = 0
\b = c
\ The cancellation law hold for R.
We now suppose that cancellation law hold in R. If possible, let a.b=0, a¹0, b¹0. We have a.b=a.0 Þb = 0, by the left cancellation law, which is a contradiction hence, the result follows.
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