Exam2Definitions

# Exam2Definitions - Math 302 Abstract Algebra Spring 2009...

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Unformatted text preview: Math 302 Abstract Algebra Spring 2009 Exam 2 Definitions 3.1 DEFINITION AND EXAMPLES OF RINGS DEFINITION • A ring is a nonempty set R together with two operations, addition, denoted + , and mul- tiplication, denoted · , that satisfy the following axioms: R1. Closure Under Addition. For all a, b ∈ R , a + b ∈ R . R2. Associative Law of Addition. For all a, b, c ∈ R , a + ( b + c ) = ( a + b ) + c. R3. Commutative Law of Addition. For all a, b ∈ R , a + b = b + a. R4. Existence of Additive Identity. There exists an element R ∈ R such that for all a ∈ R , a + 0 R = a. R5. Existence of Additive Inverses. For each a ∈ R , there exists n ∈ R such that a + n = 0 R . R6. Closure Under Multiplication. For all a, b ∈ R , a · b ∈ R . R7. Associative Law of Multiplication. For all a, b, c ∈ R , a ( bc ) = ( ab ) c. R8. Distributive Laws. For all a, b, c ∈ R , a ( b + c ) = ab + ac and ( a + b ) c = ac + bc....
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