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Boolean Rings

# Boolean Rings - The University of Oklahoma Department of...

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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 i I R i such that π k ( R ) = R k for all k I , where π k : i I R i R k is the canonical epimorphism. Example 1. The direct product 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 = 0 in R, then φ i ( r ) = 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 = 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 ) = 0. Then, by Theorem 1, Z is a subdirect product of the fields Z p . Definition 2. A ring R is subdirectly irreducible if the intersection of all nonzero ideals of R is not { 0 } .

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