Addition Modulo
Now we are going to discuss a new type of addition which is known as “addition modulo m” and written in the form
where a and b belongs to an integer and m is any fixed positive integer.By definition we have
Showing posts with label Abstract Algebra. Show all posts
Showing posts with label Abstract Algebra. Show all posts
Addition & Multiplication of Modulo - Abstract Algebra
where a and b belongs to an integer and m is any fixed positive integer.By definition we have
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Properties of Group
- The identity element of a group is always unique.
- The inverse of each element of a group is unique, i.e., in a group G with operation * for every
, there is only element
such that
, e being the identity.
Table for Group- Abstract Algebra
The composition tables are useful in examining the following axioms in the manner explained below:
- Closure Axiom : If all the elements of the table belong to the set G (say) then G is closed under the Composition a (say). If any of the elements of the table does not belong to the set, the set is not closed.
Field & Integral Domain
Integral domain:-
1-
A commutative ring with unity element and without zero divisors is called an
integral domain.
2-
An algebraic system (D,+,.) where D is a non-empty set with two binary
compositions to be denoted by addition and multiplication is called an integral
domain if following axioms are satisfied-
Definition of Ring
Ring:-
An algebraic system (R,+,.) consisting of a
non-empty set R with two binary composition (to be denoted by addition and
multiplication) is called a ring if following axiom are satisfied-
(R,+) is
an abelian group.
1- Closure axiom
2- Associative law
3- Identity element
4- Inverse axiom
(R,.) is
a semi group.
1- Closure axiom
2- Associative law
3- Multiplication distributive over
addition i.e.
a.(b+c)=a.b+a.c, ∀ a, b, c∊R
and (b+c).a=b.a+c.a, ∀a, b, c∊R
Definition of Group
An
algebraic system (G,o) where G be a non-empty set with o as defined binary
operation is called a group if following axiom are satisfied-
1- Closure Axiom:-
If a, b∊G ⇨ aob∊G , ∀ a,b∊G
then G
is said to be closed under the binary operation.
Binary Operation & Algebric Structure
Binary operation or Binary composition on a set:-
Let G be
a non-empty set then an operation ‘o’ on the non-empty set G is called binary
operation.
If a∊G, b∊G
⇒ aob∊G, ∀ a, b∊G
This
property is called closure property and if this is satisfied then G is said to
be closed under the binary composition ‘o’.
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