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Rings

Here is a clear, simple, and exam-oriented introduction to Rings from Discrete Structures / Abstract Algebra.

⭐ Introduction to Rings

A ring is an important algebraic structure in abstract algebra.
It generalizes arithmetic operations like addition and multiplication.

A ring is a set R equipped with two binary operations:

  1. Addition (+)
  2. Multiplication (·)

such that:

  • Under addition, (R, +) forms an abelian (commutative) group.
  • Under multiplication, (R, ·) is associative.
  • Multiplication distributes over addition.

In simple terms:
A ring behaves like integers without needing division.

⭐ Formal Definition of a Ring

A ring is a set (R) with two binary operations (+) and (\cdot) such that:

(A) (R, +) is an Abelian Group

  1. Closure under addition
  2. Associativity of addition
  3. Additive identity exists (0 ∈ R)
  4. Additive inverse for every element
  5. Addition is commutative

(B) (R, ·) is a Semigroup

  1. Closure under multiplication
  2. Associativity of multiplication

(C) Distributive Laws

For all (a, b, c \in R):

  1. ( a \cdot (b + c) = a \cdot b + a \cdot c )
  2. ( (a + b) \cdot c = a \cdot c + b \cdot c )

These three conditions define a ring.

⭐ Important Notes

  • A ring may or may not have a multiplicative identity (1).
  • Multiplication in a ring does not need to be commutative.

⭐ Types of Rings

1️⃣ Ring with Unity / Ring with Identity

If there exists a multiplicative identity (1).

Example:
Integers ( \mathbb{Z} ) → identity is 1

2️⃣ Commutative Ring

If multiplication is commutative:

[
a \cdot b = b \cdot a
]

Example:
Integers ( \mathbb{Z} )

3️⃣ Ring Without Identity

Rings that don’t have multiplicative identity.

Example:
Even integers 2ℤ = {…, -4, -2, 0, 2, 4, …}

4️⃣ Division Ring

Every non-zero element has a multiplicative inverse.
Multiplication need NOT be commutative.

5️⃣ Field

A commutative division ring.
Every non-zero element has a multiplicative inverse and multiplication is commutative.

Example:
Rational numbers (ℚ), Real numbers (ℝ)

⭐ Examples of Rings

✔ Example 1: Integers ℤ

Set: {…, -2, -1, 0, 1, 2, …} with usual + and ·

  • Abelian group under +
  • Associative under ·
  • Distributive laws hold
    → ℤ is a commutative ring with identity (1).

✔ Example 2: Even integers 2ℤ

  • Closed under + and ·
  • But has no multiplicative identity
    → Ring without identity.

✔ Example 3: Matrices (2×2 matrices)

Set of all 2×2 matrices with real entries.

  • Under +: Abelian group
  • Under ·: Associative
  • Distributive
    → Non-commutative ring (matrix multiplication is not commutative).

✔ Example 4: Polynomials ℝ[x]

All polynomials with real coefficients.
→ Commutative ring with identity.

⭐ Non-examples of Rings

✘ Natural numbers (ℕ)

No additive inverse → not a ring.

✘ Set with multiplication only

Missing addition → not a ring.

⭐ Summary (Exam-Friendly)

A ring R is a set with two binary operations + and · such that:

  1. (R, +) is an abelian group
  2. (R, ·) is a semigroup
  3. Multiplication distributes over addition

Rings may be:

  • With identity
  • Without identity
  • Commutative
  • Non-commutative
  • Division ring
  • Field

Examples include integers, even integers, matrices, and polynomials.