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Principal Ideal Domains

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Let \(R\) be a ring, \(I\) an ideal of \(R\). We call \(I\) Principal if it is generated by a single element. In this case, we have \(I=(g)\) where \(g\in R\) is the generator of the ideal.

We call a ring \(R\) a Principal Ideal Domain if

  1. it is an integral domain
  2. every ideal in \(R\) is a principal ideal.

Every principal ideal domain is a UFD.

If \(R\) is a PID, then \((x,y) = (\gcd(x,y))\) for any \(x,y\in R\).

1. References

  • Manuel Bronstein, Symbolic Integration 1: Transcendental Functions. Springer, second ed., 1996. Chapter 1 particularly reviews polynomial rings.
  • Nathan Jacobson, Basic Algebra I. Chapter 2.
  • Serge Lang, Algebra. Springer, third ed., GTM., 2002. Chapter 2.
  • James McIvor, Lecture Notes on Ring Theory. UC Berkeley, Math 113, Summer 2014.

Last Updated 2021-06-01 Tue 10:00.