Abstract Algebra

1. Rings
   1. Definition and Examples
      1. Basic Definition: A set equipped with two binary operations satisfying properties analogous to those of addition and multiplication.
      2. Examples of Rings:
         1. Integers under usual addition and multiplication
         2. The set of n x n matrices over a field with matrix addition and multiplication
         3. Polynomial rings over a field
         4. The ring of continuous functions from a topological space to the real numbers
         5. The ring of integers modulo n
   2. Ring Axioms
      1. Addition Operation
         1. Commutative: \(a + b = b + a\) for all \(a, b\)
         2. Associative: \(a + (b + c) = (a + b) + c\)
         3. Existence of Additive Identity: There exists an element \(0\) such that \(a + 0 = a\)
         4. Existence of Additive Inverses: For every \(a\), there exists \(-a\) such that \(a + (-a) = 0\)
      2. Multiplication Operation
         1. Associative: \(a(bc) = (ab)c\)
         2. Existence of Multiplicative Identity (in some rings): An element \(1\) such that \(a \times 1 = a = 1 \times a\)
      3. Distributive Property
         1. Left Distributive: \(a(b + c) = ab + ac\)
         2. Right Distributive: \((a + b)c = ac + bc\)
   3. Types of Rings
      1. Commutative Rings: Rings where multiplication is commutative (\(ab = ba\))
      2. Division Rings: Rings where every non-zero element has a multiplicative inverse
      3. Integral Domains: Commutative rings with no zero divisors (if \(ab = 0\), then \(a = 0\) or \(b = 0\))
         1. Fields as Extensions: Integral domains where every non-zero element is invertible
   4. Ideals and Factor Rings
      1. Principal Ideals
         1. Definition: An ideal generated by a single element
         2. Examples: \(n\mathbb{Z}\) in \(\mathbb{Z}\)
      2. Prime Ideals
         1. Definition and Properties: An ideal \(P\) such that if \(ab \in P\), then \(a \in P\) or \(b \in P\)
         2. Relation to Integral Domains: Prime ideals of the ring are crucial in making factor rings that are integral domains
      3. Maximal Ideals
         1. Definition: An ideal \(M\) such that there are no other ideals between \(M\) and the ring itself
         2. Factor Rings and Fields: The factor ring formed by a maximal ideal is a field
      4. Factorization in Rings
         1. Unique Factorization Domains: Rings where every element can be uniquely factored into irreducible elements
         2. The Role of Ideals in Factorization
   5. Ring Homomorphisms
      1. Properties and Definitions
         1. Definition: A function between two rings preserving addition and multiplication
         2. Properties: Maps identities to identities if they exist
      2. Kernel and Image
         1. Kernel as an Ideal: The set of elements mapped to zero forms an ideal
         2. Image: The set of all outputs of a homomorphism forms a subring
         3. Isomorphisms and Ring Equivalence: Ring homomorphisms that are bijections establish ring isomorphisms
   6. Polynomial Rings
      1. Construction and Properties
         1. Definition: Rings consisting of polynomials with coefficients from a particular ring
         2. Operations on Polynomial Rings: Addition and multiplication as natural extensions from coefficients
      2. Division Algorithm and Euclidean Rings
         1. Division Algorithm: The division of polynomials with remainder
         2. Euclidean Domains: Rings where Euclidean algorithm is applicable, extending the notion of divisibility to rings
      3. Irreducibility and Factorization
         1. Criteria for Irreducibility Over Fields
         2. Applications in Extensions and Field Theory