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Unformatted text preview: Partial Theorem List ( preliminary ) Jonathan L.F. King University of Florida, Gainesville FL 326112082, USA [email protected] Webpage http://www.math.ufl.edu/ ∼ squash/ 5 December, 2011 (at 00:07 ) In this document, ring means “commutative ring”. In the ring ( Z , + , · , , 1), an element u ∈ Z is a unit if ∃ w ∈ Z st. u · w = 1. ( This w is the “ multiplicative inverse ” of u , is unique, and is often written u 1 . ) In Z the units are ± 1. But in Z 12 , the ring of integers mod12, the set of units is Φ(12), i.e, {± 1 , ± 5 } . In the ring Q of rationals, each nonzero element is a unit. As a last example, in the ring G := Z + i Z of Gaussian integers , the units group is {± 1 , ± i } . ( This units group is cyclic. ) A z ∈ Z is a zerodivisor ( abbrev. ZD ) if there exists a nonzero w ∈ Z for which zw = 0. Every ring has the “ trivial zerodivisor ” —zero itself. The ring of integers doesn’t have others. In contrast, the non trivial zerodivisors of Z 12 comprise {± 2 , ± 3 , ± 4 , 6 } . Exer.E1 : In an arbitrary ring Γ, the set ZD(Γ) is disjoint from Units(Γ). Exer.E2 : Ring Z N has no irreducible elements ( see next ), since Z N is the disjointunion ZD t Units. Irreducible and prime elements. An ele ment p ∈ Z is irreducible / prime if it is neither a unit nor a zerodivisor, and : IRRED: For each factorization p = b · c , either b is a unit or c is a unit....
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 Fall '07
 JURY
 Math, Calculus

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