### Ring

An algebraic structure (R, +, .) consisting of a non-empty set R and two binary operations, called addition (+) and multiplication (.) is called a ring, if following properties are satisfied-

**1) Closure Property:**

a + b ∈ R, ∀ a , b ∈ R.

**2) Associativity Property:**

(a + b) + c = a + ( b + c ), ∀ a, b, c ∈ R.

**3) Existence of Identity:**

a + 0 = a = 0 + a, ∀ a ∈ R.

**4) Existence of Inverse:**

a + ( -a ) = 0 = ( -a ) + a.

**5) Commutative:**

a + b = b + a, ∀ a, b ∈ R.

**Types of Rings:**

**1) Null Ring:**

The binary operations addition (+) and multiplications (.) defined by ‘0+0 = 0’ and ‘0.0 = 0’ is a ring, called Null Ring or Zero Ring.

**2) Commutative Ring:**

If the multiplication in Ring is defined by a.b = b.a ∀ a , b ∈ R, than ring is known as Commutative Ring.

**3) Ring with Unity:**

If an element in Ring is defined by e.a = a.e, ∀ a, b ∈ R, than ring is known as Ring with Unity. Element e is called unit element or identity of R.

#### FIELD

A ring R with at least two elements is called a field if,

- It is commutative
- It has unity
- Each non-zero element possesses multiplicative inverse.

**Problems based on above study:**

**Prob**. Prove that a ring R is commutative, if and only if

( a + b )2 = a2 + 2ab + b2 , ∀ a, b ∈ R.

**Solution**: Click here.

**Prob**. Show that the polynomial x2 + x + 4 is irreducible over F, the field of integer modulo 11.

**Solution**:Click here.

**Prob**. If R is a ring, such that a2 = a, ∀ a ∈ R. Prove that,

1) a + a = 0, ∀ a ∈ R

2) a + b = 0 implies a = b ∀ a, b ∈ R

3) R is a commutative ring.

**Solution**:Click here.

**Prob**. Define field and prove that the set F = { 0, 1, 2, ….,6 } under addition and multiplication modulo 7 is a field.

**Solution**: Click here.

**Prob**. Define field. Prove that the set {0, 1, 2} (mod 3) is a field with respect to addition and multiplication (mod 3).

**Solution**: Click here.

**Related notes:**

- SET
- Mathematical Induction
- Relation
- Binary operations
- Algebraic struture
- Group
- Numerical problems on GROUP
- Subgroup
- Abelian Group or Commutative group
- Coset
- Factor or Quotient group
- Cyclic group
- Ring
- Numerical problems on RING
- Field
- POSET, Hasse diagram,Upper and Lower Bounds
- Hasse diagram
- Upper and Lower Bounds
- Lattice
- Recurrence relation numerical problems
- How to solve generating function

**A list of Video lectures**

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