Abstract Algebra: Groups

Rings

• A ring is a set R with two binary operations, addition (+) and multiplication (×), satisfying:
  • R is an abelian group under addition.
  • Closure under multiplication: For all a, b in R, a × b is in R.
  • Associativity of multiplication: For all a, b, and c in R, (a × b) × c = a × (b × c).
  • Distributive laws: For all a, b, and c in R, a × (b + c) = (a × b) + (a × c) and (b + c) × a = (b × a) + (c × a).
• A commutative ring is a ring in which multiplication is commutative (i.e., a × b = b × a for all a, b in R).
• A ring with unity (or identity) is a ring that contains an element 1 such that 1 × a = a × 1 = a for all a in R.
• An integral domain is a commutative ring with unity that has no zero divisors (i.e., if a × b = 0, then either a = 0 or b = 0).
• A field is a commutative ring with unity in which every non-zero element has a multiplicative inverse.
• An ideal I of a ring R is a subset of R that satisfies:
  • I is a subgroup of R under addition.
  • For all r in R and x in I, r × x and x × r are in I.
• A principal ideal is an ideal generated by a single element.
• A prime ideal P in a commutative ring R is an ideal such that if a × b is in P, then either a is in P or b is in P.
• A maximal ideal M in a ring R is an ideal such that there is no other ideal I with M ⊂ I ⊂ R.
• A factor ring (or quotient ring) is formed by the set of cosets of an ideal I in R, with the operations (a + I) + (b + I) = (a + b) + I and (a + I)(b + I) = (ab) + I.

Fields

• A field is a set F with two binary operations, addition (+) and multiplication (×), that satisfies:
  • F is an abelian group under addition, with identity element 0.
  • The non-zero elements of F form an abelian group under multiplication, with identity element 1.
  • Distributive law: For all a, b, and c in F, a × (b + c) = (a × b) + (a × c).
• Examples of fields include the rational numbers (Q), the real numbers (R), and the complex numbers (C).
• A finite field is a field with a finite number of elements.
• The characteristic of a field F is the smallest positive integer n such that n × 1 = 0, where 1 is the multiplicative identity and n × 1 means 1 + 1 + ... + 1 (n times).
  • If no such n exists, the characteristic is 0.
• Field extensions: A field E is an extension field of a field F if F is a subfield of E.
• Algebraic extensions: An element α in an extension field E of F is algebraic over F if α is a root of some non-zero polynomial with coefficients in F.
• Transcendental extensions: An element in an extension field E of F that is not algebraic over F is called transcendental over F.

Homomorphisms and Isomorphisms

• A homomorphism is a structure-preserving map between two algebraic structures (e.g., groups, rings, fields).
• Group homomorphism: A function φ: G → H between two groups (G, *) and (H, #) is a homomorphism if φ(a * b) = φ(a) # φ(b) for all a, b in G.
• Ring homomorphism: A function φ: R → S between two rings R and S is a homomorphism if φ(a + b) = φ(a) + φ(b) and φ(a × b) = φ(a) × φ(b) for all a, b in R.
• Kernel of a homomorphism: The kernel of a homomorphism φ: G → H is the set of elements in G that map to the identity element in H.
• Isomorphism: An isomorphism is a bijective homomorphism.
• If there exists an isomorphism between two algebraic structures, they are said to be isomorphic, denoted by G ≅ H.
• Isomorphic structures are essentially the same, differing only in the notation of their elements.
• Automorphism: An automorphism is an isomorphism of a mathematical object with itself.

Polynomial Rings

• A polynomial ring R[x] is the set of all polynomials in the variable x with coefficients in the ring R.
• The elements of R[x] are of the form a0 + a1x + a2x^2 + ... + anx^n, where a0, a1, ..., an are in R and n is a non-negative integer.
• Polynomial addition and multiplication are defined in the usual way.
• The degree of a polynomial is the highest power of x with a non-zero coefficient.
• Division Algorithm: For polynomials f(x) and g(x) in F[x], where F is a field and g(x) is not the zero polynomial, there exist unique polynomials q(x) and r(x) in F[x] such that f(x) = g(x)q(x) + r(x), where r(x) = 0 or deg(r(x)) < deg(g(x)).
• Irreducible polynomial: A non-constant polynomial f(x) in F[x] is irreducible over F if it cannot be factored into two non-constant polynomials in F[x].
• A root (or zero) of a polynomial f(x) is an element α such that f(α) = 0.