Basic Abstract Algebra

Hey reader! in this post we’ll discuss rings,integral domain, Fields,their various types integral domain and relations of these complicated terms with each other. Let us start with our very well known and quite simple concept ‘group’ & then sequentially discuss to bit hard concepts like Ring, integral domain,etc..

Group:

A set (Say,G) satisfying four properties below is a group:

  • Closure: ∀ a,b∈G ⇒ a.b∈G, here (.) is any function defined on G.
  • Associativity: ? a,b∈G ⇒a.(b.c) =(a.b).c
  • Identity: a.e=a, e is an identity element & e∈G
  • Inverse: a.a-1 = e, a-1 ∈ G.

These four properties constitute a group.

Cyclic Group:

If all the elements of G are powers of an element a in G, then G is cyclic .
To illustrate this, consider C={1,1,i,-i}. All the elements in C are in some powers of (i)
i=(i)1
-i=(i)3
1=(i)4

Abelian group:

Any group (G), is abelian if ∀ a,b ∈ G; a.b=b.a

Quotient Group:

If H is normal subgroup of G, then the set of all cosets of H in G is the Quotient or Factor group of G.
Symbolically ,
G/H={Ha: a∈ G}
It satisfies all group axioms.

Subgroup:

Subgroup is a subset of group, satisfying group axioms independently . In other words, subgroup is itself a group.

Cosets:

The above relation defines the cosets,
Ha={h.a/h∈H & a∈ G},
Here Ha is right coset and aH is a left coset.

Normal Subgroup:

There are two definitions of normal subgroups.
(coset form)A subgroup(H) is normal when its right coset is equal its left coset.
(general form)A subgroup(H) in G is normal if ∈ a∈G & ∀ h∈H a.h.a-1 ∈H. (i.e. a.h=h.a)

Kernel of G:

Kernel of G is the set of all elements of G which maps to identity (e’) in G’.
Symbolically ; ker f={a:f(a)=e’& a∈G}.


Now coming to some new definitions, i.e.Rings, integral domain, etc.which are inter-connected. Start with basic concept Ring:

Ring:

A non empty set R together with two binary operation +&*  then the structure (R,+,*) is ring if it satisfies

  • P1 – (R, +) is abelian with respect to + (i.e. a+b=b+a)
  • P2 – Associative with respect to * (i.e. (a*b)*c=a*(b*c) )
  • P3 – Right & left distributive laws should satisfy.

Characteristic of a Ring:

The smallest positive integer ‘n’ such that n.a =0,? a?R , then ‘n’ is the characteristic of the ring.

If ? no such positive integer then ring has zero characteristic. i.e. n.a?0 ? a?R

Boolean Ring:

A ring (R,+,*) with all of its elements idempotent( idempotent means;an=a) is the boolean ring.
i.e.  a.a = a i.e. a2=a.

Ring with unity:

A ring having multiplicative identity element, is a ring with unity.

Ideal:

A non empty subset S of a ring R is an ideal of R. If,

  • a,b ∈ S implies a-b ∈ S
  • a ∈ S and r ∈ R implies, a.r ∈ S and r.a ∈ S

Satisfy above conditions.

Zero Divisors:

Non zero elements a,b of ring R are proper or zero divisors , if a.b=0 or b.a=0
If for any ring R, a.b=0 or b.a=0 for some a or b is equal to zero, then R is the ring without zero divisors.

Integral Domain:

An integral domain ‘D’ is a commutative ring with unity containing no divisors of zero.
To be very frank,integral domain is a superset of Ring, with some additional properties.

Field:

A field ‘F’ (F,+,*) is a commutative ring with unity containing no divisors of zero & multiplicative inverse of each non zero element should exist within F (F,+,*) .

Units:

Units are the elements in a ring which have inverse under the operation multiplication.

Unique factorization domain:

An integral domain D is a unique factorization domain if, every non zero element of D is either unit or or either it is expressible as the product of finite number of prime elements of D & this factorization apart from order & associates is unique.


Here is a diagram which elaborates the relation and concepts of above terms.

Relation between terms


Hope you have read all above terms, and this post would have been helpful to you. Read our posts All about set theory & Basic topology & algebra to get detailed knowledge.