Unformatted text preview: 1.7. MODULAR ARITHMETIC 41 and .OEaŁ C OEbŁ/ C OEcŁ D OEaŁ C .OEbŁ C OEcŁ/: (b) OE0Ł is an identity element for addition; that is, for all OEaŁ 2 Z n , OE0Ł C OEaŁ D OEaŁ: (c) Every element OEaŁ of Z n has an additive inverse OE aŁ , satisfying OEaŁ C OE aŁ D OE0Ł: (d) Multiplication on Z n is commutative and associative; that is, for all OEaŁ;OEbŁ;OEcŁ 2 Z n , OEaŁOEbŁ D OEbŁOEaŁ; and .OEaŁOEbŁ/OEcŁ D OEaŁ.OEbŁOEcŁ/: (e) OE1Ł is an identity for multiplication; that is, for all OEaŁ 2 Z n , OE1ŁOEaŁ D OEaŁ: (f) The distributive law hold; that is, for all OEaŁ;OEbŁ;OEcŁ 2 Z n , OEaŁ.OEbŁ C OEcŁ/ D OEaŁOEbŁ C OEaŁOEcŁ: Multiplication in Z n has features that you might not expect. On the one hand, nonzero elements can sometimes have a zero product. For ex ample, in Z 6 , OE4ŁOE3Ł D OE12Ł D OE0Ł . We call a nonzero element OEaŁ a zero divisor if there exists a nonzero element OEbŁ such that OEaŁOEbŁ D OE0Ł . Thus, in Z 6 , OE4Ł and OE3Ł...
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 Fall '08
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 Algebra, Addition, Multiplication, one hand, additive inverse, nonzero element, Z14

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