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Homology and ringoids. I

Published online by Cambridge University Press:  24 October 2008

P. J. Hilton  and
W. Ledermann
P. J. Hilton
Affiliation:
The University Manchester
W. Ledermann
Affiliation:
The University Manchester
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Extract

In this paper we bring together two aspects of homological algebra, one modern and the other classical. MacLane(6) has suggested the desirability of avoiding superfluous references to the objects of the category in proving fundamental results in homology theory. Accordingly, we consider categories whose maps form a ringoid in the sense of Barratt(1) and impose on the ringoid axioms, expressed in the language of ring theory, but having a close affinity to the axioms of Buchsbaum(2) for an exact category. (See also Cartan and Eilenberg(3); Appendix.) The purpose of these axioms is to ensure that the standard concepts of homology theory are applicable to the ringoid. Consequently, we call such a ringoid a homological ringoid.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 54 , Issue 2 , April 1958 , pp. 152 - 167
Copyright
Copyright © Cambridge Philosophical Society 1958

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References

REFERENCES

(1)Barratt, M. G.Homotopy ringoids and homotopy groups. Quart. J. Math. (Oxford), 5 (1954), 271.Google Scholar
(2)Buchsbaum, D. A.Exact categories and duality. Trans. Amer. Math. Soc. 80 (1955), 1.Google Scholar
(3)Cartan, H. and Eilenberg, S.Homological algebra (Princeton, 1956).Google Scholar
(4)Châtelet, A.Les groupes abéliens finis (Paris, 1924).Google Scholar
(5)MacDoffee, C. C.The theory of matrices (Berlin, 1933).CrossRefGoogle Scholar
(6)MacLane, S. Review of (3), Bull. Amer. Math. Soc. 62 (1956), 615.Google Scholar