unit and unity ring theory

unit and unity ring theory

Unit and unity in ring theory, The term unit and unity should not be confused. Unity is the multiplicative identity, while a unit is any element having a multiplicative inverse. Thus, the multiplicative identity or unity is a unit, but every unit is not a unity. For example, -1 is a unit in the ring (Z, +, ×) of all integers, but -1 is not a unity, that is -1 ≠ 1

unit and unity ring theory

Commutative ring definition:

A ring in which multiplication is commutative is called a commutative ring.

Unity in ring:

A ring (R, +,  ×) with multiplicative identity 1 such that 1×a=a×1 =a for all a ∈ R is called a ring with Unity. The multiplicative identity in a ring is called a unity.

Unity ring Explanation:  unit and unity ring theory

In a ring with unity, the non-zero elements satisfy all the axioms for a group under the multiplication, except possibly the existence of multiplicative inverse. A multiplicative inverse of an element a. in a ring(R, +, ×) with unity 1 is an element a ‾ ¹  ∈ R such that a × a ‾ ¹  = a ‾ ¹× a   = 1 Since 0 ×a = a × 0 cannot have a multiplicative inverse unless we regard the set {0}, where 0 + 0 = 0 and (0)×(0) = 0 as a ring with zero as both the additive and multiplicative identity. We agree to exclude this travel case whenever we speak of a ring with unity, that is, we shall regard a ring with unity to be a nonzero ring.

Division ring:

Let (R, +,  ×) be a ring with unity. An element a ∈ R is called a unit of (R, +,  ×). If it has a multiplicative inverse in R. if every nonzero element of R is unit, Then(R, +,  ×) is called a division ring or skew field.


A commutative division ring is called a field.

Field definition:

A field (F, +, ×) is a non-empty set F having at least two elements and two binary operation + and  × (addition and multiplication) defined on F such that the following Axioms are satisfied.

(1) (F, +) is an Abelian group under addition.

(2)  (F  – {0},  ×}  is an Abelian group under multiplication.

(3)   For all a, b, c ∈  F, the right distributive law holds:

that is

(a + b) c = ac + bc

Field definition Examples