This post talks about what is Ring Theory in Mathematics also Rings and its properties in details. Rings takes the base of Group theory. Basically, a Ring is an abelian group with respect to addition. Also, it is close, Associative and distributive with respect to multiplication.
Initially, we discuss what is Ring Theory in Mathematics, later on discuss Rings and its properties.
What is Ring Theory in Mathematics:
Rings are the basic algebraic structure in Mathematics. It is the structure with two operations involving in it. They are not only addition but also multiplication. They are the backbone of various concepts, For instance, Ideals, Integral Domain, Field, etc..
In order to understand ring theory, we must be familiar with Group theory. It satisfies all the group axioms not only for addition, but also for multiplication. A little additions to this concept gives birth to new concepts such as – Fields, Ideals, Integral Domain, etc.
Now, we move towards the core concept…
Rings and its properties:
R is an algebraic structure with two operations we define as Addition (+) & Multiplication (•),we denote as R(+,•) . It satisfies certain axioms they are as below:
- R is closure under addition. ∀ a,b∈ R ⇒ (a+b) ∈ R
- R is associative under addition. ∀ a,b∈ R ⇒ (a+b)+c = a+(b+c)
- There is an identity element 0 in R such that, a+0=0, ∀ a∈R & 0∈R
- Existence of inverse element (-a) is also essential. ∀ a∈R ⇒ (-a)+a=0.
- It must be commutative under addition. (a+b)=(b+a) ∀ a,b∈ R
- R must be close under multiplication. ∀ a,b∈ R ⇒ (a•b) ∈ R
- Associativity is mandatory under multiplication.
- Distributive under multiplication is also one of its condition. i.e. ∀ a,b∈ R ⇒ a•(b+c) = a•b+a•c also, (b+c)•a = b•a+c•a
These are the essential conditions of rings. Now we move towards the properties-
There are some definitions that depend upon the ring are as follows:
Ring with unity:
A ring R with a unit element 1, such that, a.1=1.a=a , is a ring with unity.
If a R satisfies the property as- a.b= b.a ,∀ a,b∈ R. Is a commutative ring.
R is a commutative ring, then a≠0∈R is a zero divisor, if ∃ a,b ≠0 ∈ R such that a.b=0.
A commutative ring without zero divisors is an Integral domain.
If non-zero elements of a ring R form a group under multiplication is a division ring.
A field is nothing but a commutative division ring.
This is about what is Ring Theory in Mathematics also its basic properties. Next we’ll discuss about the examples and theorems on it.