Hostname: page-component-848d4c4894-hfldf Total loading time: 0 Render date: 2024-05-06T22:51:00.147Z Has data issue: false hasContentIssue false
  • English
  • Français

On the construction of ring extensions

Published online by Cambridge University Press:  18 May 2009

Edward L. Green  and
Idun Reiten
Edward L. Green
Affiliation:
Department of MathematicsUniversity of PennsylvaniaPhiladelphia 19104
Idun Reiten
Affiliation:
Department of MathematicsUniversity of Trondheim7000 TrondheimNorway
Rights & Permissions [Opens in a new window]

Extract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

Let ∧ denote a basic artin ring and r its radical. In most of this paper we assume that r2 = 0 and that Λ is a trivial extension Λ/r ⋉ r (see Section 1 for definition). Let P1 …, Pn be the non-isomorphic indecomposable projective (left) Λ-modules, and consider triples (Pi, Mi, ui), where the Mi, are (left) Λ-modules and ui:rPi → Mi/rMi isomorphisms. From this data we construct a new ring Г, which in “nice cases” has the property that r′3 = 0, Г/r′2 ≅ Λ, and r′ Qi ≅ Mi as (left) Λ-modules, where the Qi are the indecomposable projective (left) Г-modules and r′ is the radical of Г.

Type
Research Article
Information
Glasgow Mathematical Journal , Volume 17 , Issue 1 , January 1976 , pp. 1 - 11
Copyright
Copyright © Glasgow Mathematical Journal Trust 1976

References

REFERENCES

1.Auslander, M., Representation dimension of Artin algebras (Queen Mary College Notes, 1971).Google Scholar
2.Auslander, M. and Reiten, I., Notes on the representation theory of Artin algebras (Brandeis Univ., 1972).Google Scholar
3.Cartan, H. and Eilenberg, S., Homological algebra (Princeton University Press, 1956).Google Scholar
4.Eilenberg, S., Nagao, H. and Nakayama, T., On the dimension of modules and algebras IV: Dimension of residue rings of hereditary rings, Nagoya Math. J. 10 (1956), 87–95.CrossRefGoogle Scholar
5.Fossum, R., Griffith, P. and Reiten, I., Trivial extensions of abelian categories, Lecture Notes in Mathematics No. 456 (Springer-Verlag, 1975).CrossRefGoogle Scholar
6.Gabriel, P., Indecomposable representations I, Manuscr. Math. 6 (1972), 71–103.CrossRefGoogle Scholar
7.Green, E. and Gustafsen, W., Pathological Quasi-Frobenius algebras of finite type, Communications in algebra 2 (1974), 233–260.CrossRefGoogle Scholar