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nt-review

# nt-review - Partial Theorem List(preliminary Jonathan L.F...

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Partial Theorem List ( preliminary ) Jonathan L.F. King University of Florida, Gainesville FL 32611-2082, 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 , + , · , 0 , 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 mod-12, the set of units is Φ(12), i.e, 1 , ± 5 } . In the ring Q of rationals, each non-zero 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 zero-divisor ( abbrev. ZD ) if there exists a non-zero w Z for which zw = 0. Every ring has the trivial zero-divisor —zero itself. The ring of integers doesn’t have others. In contrast, the non- trivial zero-divisors 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 disjoint-union ZD t Units. Irreducible and prime elements. An ele- ment p Z is irreducible / prime if it is neither a unit nor a zero-divisor, and : IRRED: For each factorization p = b · c , either b is a unit or c is a unit.

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nt-review - Partial Theorem List(preliminary Jonathan L.F...

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