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Homology and ringoids. II

Published online by Cambridge University Press:  24 October 2008

P. J. Hilton  and
W. Ledermann
P. J. Hilton
Affiliation:
Department of MathematicsUniversity of Manchester
W. Ledermann
Affiliation:
Department of MathematicsUniversity of Manchester
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Extract

1. Introduction and recapitulation. In a previous paper (4) we have defined a ringoid (with identities) as a set of elements α, β,… in which addition and multiplication are defined for certain pairs of elements in such a way that, whenever operations are defined, the usual ring axioms are satisfied. Precisely, is an abstract category ((3), p. 108) with an addition operation such that the following axioms hold:

(R 1) For any two identity elements ι, ι′ of, let ℋ(ι, ι′) be the set of elements ofwhose left identity is ι and right identity is ι′; then ℋ(ι, ι′) is empty or an Abelian group under addition.

Information

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 55 , Issue 2 , April 1959 , pp. 149 - 164
Copyright
Copyright © Cambridge Philosophical Society 1959

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References

REFERENCES

(1)Buchsbaum, D. A.Exact categories and duality. Trans. Amer. Math. Soc. 80 (1955), 1.CrossRefGoogle Scholar
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(3)Eilenberg, S. and Steenrod, N. E.Foundations of algebraic topology (Princeton, 1952).CrossRefGoogle Scholar
(4)Hilton, P. J. and Ledermann, W.Homology and ringoids. I. Proc. Camb. Phil. Soc. 54 (1958), 152.CrossRefGoogle Scholar
(5)Hilton, P. J. and Ledermann, W.Homological ringoids. Colloq. Math. 6 (1958), 177.CrossRefGoogle Scholar
(6)Grothendieck, A.Tohoku J. Math. (1957), 119.Google Scholar