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Principal Ideal Domains and Their Modules

In: Algebra

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  • Ernest Shult

    (Kansas State University, Department of Mathematics)

  • David Surowski

Abstract

An integral domain in which every ideal is generated by a single element is called a principle ideal domain or PID. Finitely generated modules over a PID are completely classified in this chapter. They are uniquely determined by a collection of ring elements called the elementary divisors. This theory is applied to two of the most prominent PIDs in mathematics: the ring of integers, $${\mathbb Z}$$ Z , and the polynomial rings F[x], where F is a field. In the case of the integers, the theory yields a complete classification of finitely generated abelian groups. In the case of the polynomial ring one obtains a complete analysis of a linear transformation of a finite-dimensional vector space. The rational canonical form, and, by enlarging the field, the Jordan form, emerge from these invariants.

Suggested Citation

  • Ernest Shult & David Surowski, 2015. "Principal Ideal Domains and Their Modules," Springer Books, in: Algebra, edition 127, chapter 0, pages 333-354, Springer.
    • Handle: RePEc:spr:sprchp:978-3-319-19734-0_10
    DOI: 10.1007/978-3-319-19734-0_10
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