Rings in Discrete MathematicsThe ring is a type of algebraic structure (R, +, .) or (R, *, .) which is used to contain nonempty 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 nonempty set R, operation o, and operators (+ or *) on R such that:
We have some postulates that need to be satisfied. These postulates are described as follows: R1The 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:
x ∈R, y ∈R => x+y ∈ R for all x, y ∈ R
2. Association In association law, the set R will be related to composition '+' like this:
(x+y) + z = x + (y+z) for all x, y, z ∈ R.
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:
x+ y = x = 0 + x, x ∈ R
4. Existence of inverse In existence of inverse, the elements x ∈ R is exist for each x ∈ R like this:
x + (x) = 0 = (x) + x
5. Commutative of addition In the commutative law, the set R will represent for composition + like this:
x + y = y + x for all x, y ∈ R
R2Here, the set R is closed under multiplication composition like this:
xy ∈ R
R3Here, there is an association of multiplication composition like this:
(x.y).z = x.(y.z) for all x, y, z ∈ R
R4There is left and right distribution of multiplication composition with respect to addition, like this: Right distributive law
(y + z). x = y.x + z.x
Left distributive law
x.(y + z) = x.y + x.z
Types of RingThere are various types of rings, which is described as follows: Null ring 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:
0 + 0 = 0 and 0.0 = 0
Commutative ring 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:
x.y = y.x for all x, y ∈ R
The ring will be called noncommutative 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.x = x.e = x for all R
Where e can be defined as the identity of R, unity, or units elements. Ring with zero divisor If a ring contains two nonzero 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:
y.x = 0 or x.y = 0
Where 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 nonzero 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:
xy = 0 => x = 0 or y = 0
Properties of RingsAll x, y, z ∈ R if R is a ring
Next TopicOrder of Group in Discrete Mathematics
