Ring Theory Summary

Abstract Algebra · Axiom Academy

Let's review how rings add multiplicative structure to groups, and how ideals enable quotient constructions analogous to normal subgroups. Definition: A ring (R, +, ·) is an abelian group under addition with an associative multiplication that distributes over addition Two Operations: Addition forms group structure; multiplication adds algebraic richness without requiring inverses Distributive Laws: Both a(b + c) = ab + ac and (a + b)c = ac + bc must hold, linking the two operations Unity Optional: Rings with unity have multiplicative identity 1; examples like 2ℤ lack unity but are still rings Familiar Examples: ℤ, ℚ, ℝ, ℂ are rings; also ℤₙ, matrix rings Mₙ(R), and polynomial rings R[x] Subring: Subset closed under subtraction and multiplication. Examples: ℤ ⊂ ℚ ⊂ ℝ ⊂ ℂ, Gaussian integers ℤ[i] ⊂ ℂ Ideal: Special subring I with absorption property: rI ⊆ I and Ir ⊆ I for all r ∈ R Principal Ideals: In commutative rings, ideals generated by single element: ⟨a⟩ = ra | r ∈ R Maximal Ideals: Ideals M where R/M is a field; no ideals properly between M and R Prime Ideals: Ideals P where R/P is an integral domain; ab ∈ P implies a ∈ P or b ∈ P Zero Divisors: Nonzero elements a, b where ab = 0. Present in ℤₙ when n composite, absent in integral domains Units: Elements with multiplicative inverses. Unit group U(R) consists of all invertible elements In ℤₙ: Element a is unit iff gcd(a, n) = 1; element a is zero divisor iff 1 < gcd(a, n) < n

This is the written version of the interactive lesson above. See the full Abstract Algebra course.