Hostname: page-component-848d4c4894-5nwft Total loading time: 0 Render date: 2024-05-30T01:14:14.593Z Has data issue: false hasContentIssue false

# 3 - Factorization in semifirs

Published online by Cambridge University Press:  22 August 2009

## Summary

For the study of non-commutative unique factorization domains we begin by looking at the lattice of factors and the notion of similarity for matrices in Section 3.1. The resulting concept of non-commutative UFD, in Section 3.2, is mainly of interest for the factorization of full n×n matrices over 2n-firs; thus it can be applied to study factorization in free algebras. Another class, the rigid UFDs, forming the subject of Section 3.3, generalizes valuation rings and is exemplified by free power series rings. We also examine various direct decomposition theorems (Sections 3.4 and 3.5), but throughout this chapter we only consider square (full) matrices, corresponding to torsion modules over semifirs. The factorization of rectangular matrices, which is much less well developed, will be taken up in Chapter 5.

Similarity in semifirs

To study factorizations in non-commutative integral domains it is necessary to consider modules of the form R/aR. We recall from Section 1.3 that two right ideals a, a′ of a ring R are similar if R/a ≅ R/a′. In the case of principal ideals the similarity of aR and a′R (for regular elements a and a′) just corresponds to the similarity of the elements a and a′ as defined in Section 0.5 (see Proposition 0.5.2 and the preceding discussion, as well as Section 1.3).

In a semifir it is possible to simplify this condition still further.

Type
Chapter
Information
Publisher: Cambridge University Press
Print publication year: 2006

## Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

# Save book to Kindle

Note you can select to save to either the @free.kindle.com or @kindle.com variations. ‘@free.kindle.com’ emails are free but can only be saved to your device when it is connected to wi-fi. ‘@kindle.com’ emails can be delivered even when you are not connected to wi-fi, but note that service fees apply.

Find out more about the Kindle Personal Document Service.

• Factorization in semifirs
• Book: Free Ideal Rings and Localization in General Rings
• Online publication: 22 August 2009
• Chapter DOI: https://doi.org/10.1017/CBO9780511542794.008
Available formats
×

# Save book to Dropbox

To save content items to your account, please confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your account. Find out more about saving content to Dropbox.

• Factorization in semifirs
• Book: Free Ideal Rings and Localization in General Rings
• Online publication: 22 August 2009
• Chapter DOI: https://doi.org/10.1017/CBO9780511542794.008
Available formats
×

# Save book to Google Drive

To save content items to your account, please confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your account. Find out more about saving content to Google Drive.

• Factorization in semifirs
• Book: Free Ideal Rings and Localization in General Rings
• Online publication: 22 August 2009
• Chapter DOI: https://doi.org/10.1017/CBO9780511542794.008
Available formats
×