Introduction to Abstract Algebra Notes

Course Overview
- Lecture notes prepared by Stephen Edward Moore, University of Cape Coast, Ghana.
- Includes contributions from Mr. Solomon Nortey.

Importance of Definitions
- Understand meanings, clauses, and conditions of mathematical concepts.
- Example: Understanding injective mapping is crucial for proofs related to it.
Studying Theorems
- Memorization of theorems is unnecessary; understanding proofs is key.
- Summarizing key ideas of proofs aids in memory and comprehension.

Cumulative Nature
- Material becomes more complex over time; requires constant engagement.
- Ask questions about definitions and their implications.

MAPPING
- Definitions: Domain, Co-Domain, Range, Equal, Injective, Surjective, Identity, Bijective, Product, Invertible Mappings.
BINARY OPERATION
- Properties: Closure, Associativity, Commutativity, Identity and Inverses.
RELATIONS
- Types: Empty, Reflexive, Symmetric, Transitive, Antisymmetric, Identity, Inverse, Universal.
MATHEMATICAL INDUCTION
- Technique with two steps: base case and inductive step.
DIVISORS
- Definitions, Greatest Common Divisor, Division Algorithm, and Euclidean Algorithm.
CONGRUENCE MODULO N
- Definitions, Properties, Operations, Linear Congruences, Chinese Remainder Theorem.
RATIONAL NUMBERS
- Definitions, Operations, Properties, Decimal Representations.
COMPLEX NUMBERS
- Definitions, Operations, Trigonometric Forms, nth Roots, and De Moivre's Theorem.
GROUPS
- Definitions, Properties, Abelian Groups, Integral Exponent and Uniqueness theorems.
RINGS
- Definitions of rings, properties, and multiplicative inverses.
FIELDS
- Definitions, conditions for fields, and specific examples with Zp.

Engage actively with the material, ask questions, and explore examples.
Encourage collaboration for peer learning, especially with exercises and problems.