Exam2Theorems - Math 302 Exam 2 Theorems Chapter 3 Rings...

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Math 302 Abstract Algebra Spring 2009 Exam 2 Theorems Chapter 3 Rings 3.2 BASIC PROPERTIES OF RINGS THEOREM 3.3 For any element a in a ring R , the equation a + x = 0 R has a unique solution. That is, for each a R , there exists a unique n R such that a + n = 0 R . THEOREM 3.4 (CANCELLATION PROPERTY) Let R be a ring. For all a, b, c R , if a + b = a + c , then b = c . THEOREM 3.5 For any elements a and b of a ring R , (1) a · 0 R = 0 R = 0 R · a . (2) a ( b ) = ( ab ) = ( a ) b . (3) ( a ) = a . (4) ( a + b ) = ( a ) + ( b ) . (5) ( a b ) = a + b . (6) ( a )( b ) = ab . If R has an identity, then for any a R (7) ( 1 R ) a = a . THEOREM 3.6 Let R be a ring and S be a nonempty subset of R that satis±es the following properties: SR1. Closure Under Subtraction. For all a, b S , a b S . SR2. Closure Under Multiplication. For all a, b S , ab S . Then S is a subring of R . THEOREM 3.9 Every ±eld F is an integral domain. THEOREM 3.10 Cancellation is valid in any integral domain R . Suppose R is an integral domain. For all a, b, c R , a 6 = 0 R , if ab = ac , then b = c . THEOREM 3.11 Every ±nite integral domain is a ±eld. 3.3 ISOMORPHISMS AND HOMOMORPHISMS THEOREM 3.12 Let f : R S be a ring homomorphism. Then (1) f (0 R ) = 0 S . (2)
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This note was uploaded on 10/20/2009 for the course MATH 302 taught by Professor Edwards during the Spring '03 term at CSU Fullerton.

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Exam2Theorems - Math 302 Exam 2 Theorems Chapter 3 Rings...

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