Zero Divisors and Units

Abstract Algebra · Axiom Academy

LESSON Zero Divisors and Units Understanding special elements in rings: those that multiply to zero, and those with multiplicative inverses 1. Zero Divisors: Unexpected Zeros This is surprising because in familiar systems like ℤ or ℝ, if a product equals zero, at least one factor must be zero. But in modular arithmetic Z n , this isn't always true! In Z n , zero divisors arise from divisors of n. If d divides n (with 1 < d < n), then d is a zero divisor because there exists another divisor d' where d · d' ≡ 0 (mod n). 3. Units: Elements with Inverses In Z n , an element a is a unit if and only if gcd(a, n) = 1. These are the elements that are coprime to n—they have no common factors with n. The set of all units in a ring R forms a group under multiplication, denoted U(R) or R×. For Z n , we denote this group as U(Z n ) or simply U(n). Closed: Product of two units is a unit Inverses: Every unit has a multiplicative inverse Size: |U(n)| = φ(n), where φ is Euler's totient function

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