Boolean Rings

Boolean Rings - The University of Oklahoma Department of Mathematics Speaker Houssein El Turkey Student Algebra Seminar Date February 5th 2009

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Unformatted text preview: The University of Oklahoma Department of Mathematics Speaker: Houssein El Turkey Student Algebra Seminar Date: February 5th, 2009 Boolean rings In this talk, we are going to define Boolean rings and study their structure. 1 preliminaries Definition 1. A ring R is said to be a subdirect product (or sum) of the family of rings { R i | i ∈ I } if R is a subring of the direct product Q i ∈ I R i such that π k ( R ) = R k for all k ∈ I , where π k : Q i ∈ I R i → R k is the canonical epimorphism. Example 1. The direct product Q i ∈ I R i is itself a subdirect product of the rings R i . There could be other subdirect products of the rings R i . If a ring R is isomorphic to a subdirect product T of rings R i , i ∈ I , T may be called a representation of R as a subdirect product of the rings R i . Theorem 1. A ring R has a representation as a subdirect product of rings R i , i ∈ I iff for each i ∈ I , there exists an epimorphism φ i : R → R i such that if r 6 = 0 in R, then φ i ( r ) 6 = 0 for at least one i ∈ I . Example 2. The ring Z is a subdirect product of the fields Z p for all prime numbers p . To prove this, let φ p : Z → Z p be the canonical epimorphism φ p ( n ) = n ( modp ) = [ n ] p . If r 6 = 0 in Z , then r can not be a multiple of all primes p ∈ Z , and hence there is at least one prime number p such that φ p ( r ) 6 = 0. Then, by Theorem 1, Z is a subdirect product of the fields Z p ....
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This note was uploaded on 01/26/2010 for the course MATH-M 413 taught by Professor Michaeljolly during the Spring '08 term at Indiana.

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Boolean Rings - The University of Oklahoma Department of Mathematics Speaker Houssein El Turkey Student Algebra Seminar Date February 5th 2009

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