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Asymmetry in the lattice of kernel functors

Published online by Cambridge University Press:  18 May 2009

Ana M. de Viola-Prioli  and
Jorge E. Viola-Prioli
Ana M. de Viola-Prioli
Affiliation:
Universidad Simón Bolívar, Caracas, Venezuela
Jorge E. Viola-Prioli
Affiliation:
Universidad Simón Bolívar, Caracas, Venezuela
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Much of the research done by different authors on the lattice of kernel functors (equivalently, linear topologies) has been summarized by Golan in [2]. More recently, the rings whose lattices of kernel functors are linearly ordered were introduced in [3] as a categorical generalization of valuation rings in the non-commutative case. Results (and examples) in [3] show that there is an abundance of non-commutative rings R whose lattices (R), both in Mod-R and R-Mod, are simultaneously linearly ordered; however, the question of the symmetry of this condition remained open. Here we will prove that, for every natural number n≥3, there exists a ring Rn such that (Mod-Rn) is a linearly ordered lattice of n elements, whereas (Rn-Mod) is not linearly ordered.

Type
Research Article
Information
Glasgow Mathematical Journal , Volume 33 , Issue 1 , January 1991 , pp. 95 - 97
Copyright
Copyright © Glasgow Mathematical Journal Trust 1991

References

REFERENCES

1.Cohn, P. M., Free rings and their relations (Academic Press, 1971).Google Scholar
2.Golan, J. S., Linear topologies on a ring: an overview, Research Notes in Mathematics No. 159 (Pitman, 1987).Google Scholar
3.de Viola-Prioli, A. M. and Viola-Prioli, J., Rings whose kernel functors are linearly ordered, Pacific J. Math. 132 (1988), 21–34.CrossRefGoogle Scholar