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Rings of a Set
A non-empty set R together with two binary operations ‘*’ and Δ, defined in it, is said to form a ring if the following axioms are satisfied.

(a) Axioms due to the composition “*”:

R1: R is closed w.r.t. “*”, i.e. ∀ a, b R a * b R.

R2: * is associative in R i.e. a * (b * c) = (a * b) * c, ∀ a, b, c R.

R3: Identity element w.r.t. *. There exists e R such that

e * a = a * e = a, for every a R, then

R4: Existence of inverse of every element:
For every a R, there exists b R such that a * b = b * a = e, then b is called the inverse of a and vice versa.

R5: * is commutative in R, i.e. a * b = b * a. ∀ a, b R.

(b) Axioms due to both the compositions * a and Δ.

R6: R is closed w.r.t. Δ, i.e. ∀ a, b R a Δ b R.

R7: Δ is associative in R, i.e. a Δ (b Δ c) = (a Δ b) Δ c, ∀ a, b, c R.

R8: Distributive Laws:

Left distributive law: a Δ (b Δ c) = (a Δ b)* (a Δ c), ∀ a, b, c R.

Right distributive law: (b * c) Δ a = (b Δ a)*(c Δ a), ∀ a, b, c R.

In other words, an algebraic structure < R, *, Δ > is said to form an associative ring if

(a) < R, * > is an algebraic group

(b) < R, Δ > is a semi-group

(c) Both right and left distributive laws hold in R.

Some special types of rings

(I) Ring with unity: A ring < R, * Δ > is said to be a ring with unity, if there exists an element g R such that

g Δ a = a = a Δ g                     ∀ a R   

then the element g is called the unity of the ring.

(II) Commutative ring: A ring < R, *, Δ > is said to be a commutative ring if Δ is commutative in R

i.e. a Δ b = b Δ a                       ∀ a, b R   

(III) Ring without zero divisors: A ring < R, *, Δ > is said to be ring without zero divisors if the product of two non-zero elements or R is zero, i.e. if ∀ a, b R,

a Δ b = 0 either a = 0 or b = 0.

(IV) Ring with zero divisors:
A ring < R, *, Δ > is said to be ring with zero divisors if for some pair of elements ∀ a, b R, a ≠ 0, b ≠ 0, but, we have

a Δ b = 0.

The ring of integers with addition and multiplication module 5 is a ring with zero divisors as 3 × 57 = 0 so 3 and 7 are zero divisors.

(V) Integral domain: A ring < R, *, Δ > which is such that

(i) it is commutative, (ii) it has a unity, (iii) it is without zero divisors, is called an integral domain.

(VI) Division ring or skew field: A triplet < R, *, Δ > is said to be skew-field or division field if

(i) < R, *, Δ > is a ring with unity

(ii) For all a R ~ (0), there exists an element bR ~ (0) such that

a Δ b = b Δ a = g

In other words a R ~ (0) is invertible.

(VII) Field: A commutative division ring is called a field.

(VIII) Idempotent ring: A ring R is said to be idempotent if

x2 = x, ∀ x R

(IX) Boolean ring: A ring R is said to be Boolean ring if for every x R, x2 = x, i.e. if the ring is idempotent.

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