Rings in Discrete Mathematics
The ring is a type of algebraic structure (R, +, .) or (R, *, .) which is used to contain non-empty set R. Sometimes, we represent R as a ring. It usually contains two binary operations that are multiplication and addition.
An algebraic system is used to contain a non-empty set R, operation o, and operators (+ or *) on R such that:
- (R, 0) will be a semigroup, and (R, *) will be an algebraic group.
- The operation o will be said a ring if it is distributive over operator *.
We have some postulates that need to be satisfied. These postulates are described as follows:
The algebraic group is described by the system (R, +). So it contains some properties, which is described as follows:
1. Closure Property
In the closure property, the set R will be called for composition ‘+’ like this:
In association law, the set R will be related to composition ‘+’ like this:
3. Existence of identity
Here, R is used to contain an additive identity element. That element is known as zero elements, and it is denoted by 0. The syntax to represent this is described as follows:
4. Existence of inverse
In existence of inverse, the elements x ∈ R is exist for each x ∈ R like this:
5. Commutative of addition
In the commutative law, the set R will represent for composition + like this:
Here, the set R is closed under multiplication composition like this:
Here, there is an association of multiplication composition like this:
There is left and right distribution of multiplication composition with respect to addition, like this:
Right distributive law
Left distributive law
Types of Ring
There are various types of rings, which is described as follows:
A ring will be called a zero ring or null ring if singleton (0) is using with the binary operator (+ or *). The null ring can be described as follows:
The ring R will be called a commutative ring if multiplication in a ring is also a commutative, which means x is the right divisor of zero as well as the left divisor of zero. The commutative ring can be described as follows:
The ring will be called non-commutative ring if multiplication in a ring is not commutative.
Ring with unity
The ring will be called the ring of unity if a ring has an element e like this:
e can be defined as the identity of R, unity, or units elements.
Ring with zero divisor
If a ring contains two non-zero elements x, y ∈ R, then the ring will be known as the divisor of zero. The ring with zero divisors can be described as follows:
x and y can be said as the proper divisor of zero because in the first case, x is the right divisor of zero, and in the second case, x is the left divisor of zero.
0 is described as additive identity in R
Ring without zero divisor
If products of no two non-zero elements is zero in a ring, the ring will be called a ring without zero divisors. The ring without zero elements can be described as follows:
Properties of Rings
All x, y, z ∈ R if R is a ring
- (-x)(-y) = xy
- x0 = 0x = 0
- (y-z)x = yx- zx
- x(-y) = -(xy) = (-x)y
- x(y-z) = xy – xz