Principal Ideal Domains and Their Modules

Ernest Shult () and David Surowski
Additional contact information
Ernest Shult: Kansas State University, Department of Mathematics

Chapter Chapter 10 in Algebra, 2015, pp 333-354 from Springer

Abstract: 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.

Keywords: Principal Ideal Domain (PID); Rational Canonical Form; Bezout Domain; Invariant Factor Theorem; Minimal Polynomial (search for similar items in EconPapers)
Date: 2015
References: Add references at CitEc
Citations:

There are no downloads for this item, see the EconPapers FAQ for hints about obtaining it.

Related works:
This item may be available elsewhere in EconPapers: Search for items with the same title.

Export reference: BibTeX RIS (EndNote, ProCite, RefMan) HTML/Text

Persistent link: https://EconPapers.repec.org/RePEc:spr:sprchp:978-3-319-19734-0_10

Ordering information: This item can be ordered from
http://www.springer.com/9783319197340

DOI: 10.1007/978-3-319-19734-0_10

Access Statistics for this chapter

More chapters in Springer Books from Springer
Bibliographic data for series maintained by Sonal Shukla () and Springer Nature Abstracting and Indexing ().