Copyright
Jan E. GrabowskiPublished On
2025-09-22Page Range
pp. 147–194Language
- English
Print Length
48 pages
5. Examples
- Jan E. Grabowski (author)
In this chapter, we will use the general results of the previous one in various different situations. We start with the case of principal ideal domains and use the ideas we have seen to give a classification of finitely generated modules over such rings. As a special case, this yields the classification of finitely generated Abelian groups.
Next, we turn back to groups. We introduce an algebra, the group algebra, whose representation theory is the same (in a strong formal sense) as that of the group. Thirdly, we study the representation theory of quivers in terms of modules in a similar way. Again, we can introduce an algebra, the path algebra, whose representation theory is the same as that of the quiver.
Contributors
Jan E. Grabowski
(author)Jan Grabowski is an experienced researcher and educator in mathematics, with over 20 years’ experience of both, and currently holds the position of Professor of Algebra at Lancaster University. Jan has a strong research track record of publications in algebra and related topics, both as a single author and collaboratively. He has taught courses at Oxford and Lancaster across a range of levels and has been teaching abstract algebra for the majority of this time, including covering aspects of the material in the book in courses at both institutions. Jan has had recognition for his teaching, including a teaching prize at Oxford, obtaining a Postgraduate Certificate in Academic Practice and being awarded Senior Fellowship of the Higher Education Academy. He has championed innovation in teaching by presenting a mature exposition of ‘how mathematicians really think about these things’ as compared with other approaches that often defer more advanced ideas or techniques, rather than encouraging students to engage with challenging ideas early and repeatedly.