Rings with Chain Conditions

Mahima Ranjan Adhikari and Avishek Adhikari
Additional contact information
Mahima Ranjan Adhikari: Institute for Mathematics, Bioinformatics, Information Technology and Computer Science (IMBIC)
Avishek Adhikari: University of Calcutta, Department of Pure Mathematics

Chapter Chapter 7 in Basic Modern Algebra with Applications, 2014, pp 257-271 from Springer

Abstract: Abstract Chapter 7 continues to develop the theory of rings and studies chain conditions for ideals of a ring. The motivation came from an interesting property of the ring of integers Z that its every ascending chain of ideals terminates. This interesting property of Z was first recognized by the German mathematician Emmy Noether (1882–1935). This property leads to the concept of Noetherian rings. On the other hand, Emil Artin (1898–1962) showed that there are some rings in which every infinite descending chain of ideals terminates. Such rings are called Artinian rings. The ring Z has an infinite descending chain of ideals. This property of Z shows that Z is not an Artinian ring. In this chapter special classes of rings are also studied and deeper results on ideal theory are proved. Some finiteness condition for ideals is imposed and Noetherian rings are introduced which are versatile. The most convenient equivalent formulation of the Noetherian requirement is that the ideals of the ring satisfy the ascending chain condition. A connection between a Noetherian domain and a factorization domain is established. Moreover, the Hilbert Basis Theorem is proved. This theorem has made a revolution and has given an extensive class of Noetherian rings. Its application to algebraic geometry is also discussed. The study in this chapter culminates in rings with a descending chain condition for ideals, which determines their ideal structure.

Keywords: Descending Chain Condition; Hilbert Basis Theorem; Artinian Ring; Convenient Equivalent Formulation; Noetherian Ring (search for similar items in EconPapers)
Date: 2014
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:978-81-322-1599-8_7

Ordering information: This item can be ordered from
http://www.springer.com/9788132215998

DOI: 10.1007/978-81-322-1599-8_7

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 ().