**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**

**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.**

**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**