Abstract Algebra

Abstract Algebra

Meaning

• Abstract Algebra studies algebraic structures
• Focus: sets + operations and their properties (rules)

Algebraic Structure

• An algebraic structure is: (S, operations)
• Example:
  • $(ℤ, +)$ integers under addition
  • $(ℕ, +, ×)$ naturals under addition and multiplication

Binary Operation

• A binary operation on set $S$: $*: S × S → S$
• Properties often checked:
  1. Closure: $a*b ∈ S$
  2. Associativity: $(a_b)c = a(b_c)$
  3. Commutativity: $a_b = b_a$
  4. Identity: $e_a = a_e = a$
  5. Inverse: $a_a⁻¹ = a⁻¹_a = e$
  6. Distributive: $a*(b+c) = a_b + a_c$

Algebraic Structures
│
├─ Magma (set + binary operation)
│  │
│  └─ Semigroup (associative)
│     │
│     └─ Monoid (has identity)
│        │
│        └─ Group (has inverses)
│           │
│           ├─ Abelian Group (commutative)
│           │
│           └─ Non-Abelian Group
│
│
├─ Ring (two operations: +, ×)
│  │
│  ├─ Commutative Ring
│  │  │
│  │  ├─ Integral Domain (no zero divisors)
│  │  │  │
│  │  │  └─ Field (division possible)
│  │  │
│  │  └─ Ring with Zero Divisors
│  │
│  └─ Non-Commutative Ring
│     │
│     ├─ Division Ring (non-commutative field)
│     │
│     └─ Matrix Ring
│
│
└─ Vector Space (over a field)
   │
   └─ Algebra (vector space + multiplication)

1. Semigroup

A semigroup $(S, *)$:

• Closure
• Associativity

Examples:

• $(ℕ, +)$
• $(ℕ, ×)$
• $(ℤ, +)$
• $(ℤ, ×)$
• $(S, \text{max})$, $(S, \text{min})$

1.1 Monoid

A monoid $(M, *)$:

• Semigroup
• Identity element exists

Examples:

• $(ℕ, +)$, identity $0$
• $(ℕ, ×)$, identity $1$
• $(ℤ, +)$, identity $0$
• $(ℤ, ×)$, identity $1$
• $(P(S), ∪)$, identity $∅$
• $(P(S), ∩)$, identity $S$

1.1.1 Group

A group $(G, *)$:

• Monoid
• Inverse exists for every element

Examples:

• $(ℤ, +)$
• $(ℚ, +)$
• $(ℝ, +)$
• $(ℚ\setminus{0}, ×)$
• $(ℝ\setminus{0}, ×)$

1.2 Abelian Monoid (Commutative)

An Abelian Monoid:

• Monoid
• Commutative

Examples:

• $(ℕ, +)$
• $(ℕ, ×)$
• $(ℤ, +)$
• $(P(S), ∪)$
• $(P(S), ∩)$

1.2.1 Abelian Group (Commutative)

An Abelian Group:

• Group
• Commutative

Examples:

• $(ℤ, +)$
• $(ℚ, +)$
• $(ℝ, +)$
• $(ℚ\setminus{0}, ×)$
• $(ℝ\setminus{0}, ×)$

2. Ring

A ring $(R, +, ×)$ satisfies:

1. $(R, +)$ is an abelian group
2. $(R, ×)$ is a semigroup
3. Distributive laws:
   • $a×(b+c) = a×b + a×c$
   • $(a+b)×c = a×c + b×c$

Important correction:

• Ring being an abelian group refers only to addition
• Multiplication is independent of addition

Examples:

• $(ℤ, +, ×)$
• $(ℚ, +, ×)$
• $(ℝ, +, ×)$
• $(M_n(ℝ), +, ×)$ (non-commutative ring)

2.1 Commutative Ring

A commutative ring:

• Ring
• Multiplication is commutative

Clarification (important):

• Ring is abelian only under addition
• Multiplication need not be commutative
• Hence commutative ring is defined separately

Examples:

• $(ℤ, +, ×)$
• $(ℚ, +, ×)$
• $(ℝ, +, ×)$

Non-example:

• $(M_n(ℝ), +, ×)$ is a ring but not commutative

2.2 Ring with Identity (Unity)

A ring with identity:

• Ring
• Multiplicative identity $1$ exists
• $1×a = a×1 = a$

Clarification (important):

• Ring axioms do not require multiplicative identity
• Semigroup under multiplication does not guarantee identity
• Hence defined separately

Examples:

• $(ℤ, +, ×)$
• $(ℚ, +, ×)$
• $(ℝ, +, ×)$
• Polynomial ring $(ℝ[x], +, ×)$

2.1.1 Integral Domain

A commutative ring with unity such that:

• No zero divisors

Meaning: If $a×b = 0$ then $a=0$ or $b=0$

Example:

• $(ℤ, +, ×)$ integral domain

2.1.1.1 Field

A field $(F, +, ×)$ satisfies:

1. $(F, +)$ is abelian group
2. $(F − {0}, ×)$ is abelian group
3. Distributive laws

Example:

• $(ℚ, +, ×)$ field
• $(ℝ, +, ×)$ field

3. Vector Space

A vector space is built over a field:

$(V, +)$ is abelian group

Scalar multiplication: $F × V → V$

Example:

• $ℝ^n$ over field $ℝ$

Lattice (Important in Discrete Math)

A structure $(L, ∨, ∧)$ satisfying:

• Idempotent, commutative, associative
• Absorption laws

Example:

• $(P(S), ∪, ∩)$ power set forms lattice

Homomorphism (Structure Preserving Map)

A function $f: A → B$ that preserves operation:

For groups: $f(a*b) = f(a) * f(b)$

For rings:

• $f(a+b)=f(a)+f(b)$
• $f(a×b)=f(a)×f(b)$

Isomorphism

A bijective homomorphism

• Means two structures are “same” algebraically

17) Quick Summary Table

	Structure	Operations	Key Condition
1	Semigroup	$*$	closure + associative
2	Monoid	$*$	semigroup + ==identity==
3	Group	$*$	monoid + inverse
4	Abelian Group	$*$	group + commutative
5	Ring	$+,×$	($+$) abelian group + distributive
6	Field	$+,×$	ring + every nonzero has inverse

---

Abelian Group & Abelian Monoid

Basic Idea

• These are algebraic structures: a set + an operation
• Operation is usually written as: * or +

Binary Operation

• Let $S$ be a set. A binary operation on $S$ is a function: $*: S × S → S$
• Meaning: for any $a, b ∈ S$, result $a * b ∈ S$

1. Semigroup

A semigroup is $(S, *)$ such that:

1. Closure: $a*b ∈ S$
2. Associative: $(a_b)c = a(b_c)$ for all $a,b,c ∈ S$

2. Monoid

A monoid is a semigroup + identity. A monoid $(M, *)$ satisfies:

1. Closure
2. Associativity

3. Identity element exists

   • There exists $e ∈ M$ such that:
   • $e_a = a_e = a$ for all $a ∈ M$

Example:

• $(ℕ, +)$ is a monoid
  • identity = $0$

3. Abelian Monoid (Commutative Monoid)

An abelian monoid is a monoid + commutativity.

So $(M, *)$ satisfies:

1. Closure
2. Associativity

3. Identity exists

4. Commutative: $a_b = b_a$ for all $a,b ∈ M$

Example:

• $(ℕ, +)$ is abelian monoid (identity = 0)
• $(ℕ, ×)$ is also abelian monoid (identity = 1)

4. Group

A group $(G, *)$ is a monoid + inverse for every element.

It satisfies:

1. Closure
2. Associativity
3. Identity exists
4. Inverse exists
   • For every $a ∈ G$, there exists $a⁻¹ ∈ G$ such that:
   • $a_a⁻¹ = a⁻¹_a = e$

Example:

• $(ℤ, +)$ is a group
  • identity = $0$
  • inverse of $a$ is $-a$

5. Abelian Group (Commutative Group)

An abelian group is a group + commutativity.

So $(G, *)$ satisfies:

1. Closure
2. Associativity
3. Identity exists
4. Inverse exists
5. Commutative: $a_b = b_a$

Example:

• $(ℤ, +)$ abelian group
• $(ℚ, +)$ abelian group
• $(ℝ, +)$ abelian group

Key Difference (Monoid vs Group)

• Monoid: identity exists, but inverse may not exist

• Group: identity exists + every element has inverse

Example:

• $(ℕ, +)$ is monoid but not group (no inverse for 5 in ℕ)
• $(ℤ, +)$ is group (inverse for 5 is -5 in $ℤ$)

Structure	Closure	Associative	Identity	Inverse	Commutative	ℕ Operations	$ℤ$ Operations
Semigroup	Yes	Yes	No	No	Not necessary	+ , ×, max, min	+ , ×, max, min
Monoid	Yes	Yes	==Yes==	No	Not necessary	+(0), ×(1), max(0), min((∞)	+(0), ×(1)
==Abelian== Monoid	Yes	Yes	==Yes==	No	==Yes==	+ , ×, max, min	+ , ×, max, min
Group	Yes	Yes	==Yes==	==Yes==	Not necessary	❌	+
==Abelian== Group	Yes	Yes	==Yes==	==Yes==	==Yes==	❌	+

Key Points

• ℕ is not a group (no inverse)

• (ℤ, +) is Abelian group

• (ℤ, ×) is not a group (no inverse for 2)

• − and / are not binary operations on ℤ or ℕ

To Memorize ⭐⭐

• Closure + Associativity → present in all algebraic structures
• Semigroup → Closure + Associativity

• Monoid== → Semigroup + ==Identity

• Group== → Monoid + ==Inverse

• Abelian → Commutative