Finding All Ideals of Z₁₂

Abstract Algebra · Axiom Academy

EXAMPLE Finding All Ideals of Z₁₂ Discover the beautiful correspondence between ideals and divisors in modular arithmetic Excellent work! You've discovered all ideals of and their correspondence with divisors. Remember: Divisor Correspondence: Every ideal of corresponds to a unique divisor d of n , and vice versa Generator Formula: For divisor d of n , the ideal is containing all multiples of n/d Number of Ideals: has exactly as many ideals as n has divisors Size Pattern: If d divides n , then (the ideal has d elements) Inclusion: For divisors d_1 | d_2 , we have n/d_2 n/d_1 Extremes: The trivial ideal corresponds to d=1 , and itself corresponds to d=n This beautiful theorem extends to all ! Try finding all ideals of or using the same method: find divisors, then construct ideals for each divisor d .

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