{[ promptMessage ]}

Bookmark it

{[ promptMessage ]}

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

Info icon This preview shows pages 1–2. Sign up to view the full content.

View Full Document Right Arrow Icon
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 } .
Image of page 1

Info icon This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document Right Arrow Icon
Image of page 2
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}

What students are saying

  • Left Quote Icon

    As a current student on this bumpy collegiate pathway, I stumbled upon Course Hero, where I can find study resources for nearly all my courses, get online help from tutors 24/7, and even share my old projects, papers, and lecture notes with other students.

    Student Picture

    Kiran Temple University Fox School of Business ‘17, Course Hero Intern

  • Left Quote Icon

    I cannot even describe how much Course Hero helped me this summer. It’s truly become something I can always rely on and help me. In the end, I was not only able to survive summer classes, but I was able to thrive thanks to Course Hero.

    Student Picture

    Dana University of Pennsylvania ‘17, Course Hero Intern

  • Left Quote Icon

    The ability to access any university’s resources through Course Hero proved invaluable in my case. I was behind on Tulane coursework and actually used UCLA’s materials to help me move forward and get everything together on time.

    Student Picture

    Jill Tulane University ‘16, Course Hero Intern