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concepts from set theory the system of natural numbers
SECTION PAGE 1 Operations on sets
Product sets mappings
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additive group arbitrary ascending chain condition associated primes automorphism binary composition called cancellation law clear coefficients commutative group complex numbers composition series consider correspondence coset cyclic group decomposition define definition denote determined direct product division ring divisor elements of 21 endomorphism example EXERCISES exists factor group field finite number fl a2 following Theorem function Gaussian group of order groups with operators Hence holds ideal in 21 implies indecomposable integral domain intersection invariant M-subgroups inverse irreducible elements irredundant isomorphism kernel left ideal Lemma M-group matrix modular lattice module morphism natural numbers Noetherian ring non-zero elements obtain partially ordered set permutation pi U p2 polynomial positive integers primary ideals principal ideal domain Proof prove the following rational numbers real numbers relative result ring 21 satisfies semi-group Show subgroup submodule subring subset suppose theory tion transcendental transformation group uniqueness vector verify zero-divisor