Density relative to a torsion theory

Bland, Paul; Riley, Stephen

doi:10.1090/S0002-9939-1981-0614872-7

Density relative to a torsion theory
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by Paul Bland and Stephen Riley PDF

Proc. Amer. Math. Soc. 82 (1981), 527-532 Request permission

Abstract:

If $(\Im , \mathcal {F})$ is a torsion theory on Mod $R$, then a ring $B$ of biendomorphisms of a $\Im$-cocritical module is topologized. Moreover, if a certain factor module of $R$ is quasi-projective, a ring monomorphism $\varphi :R \to B$ is found such that $\varphi (R)$ is topologically dense in $B$. This is done in such a way that when $(\Im , \mathcal {F})$ is the torsion theory in which every module is torsion free, the Jacobson density theorem is recovered.

References

Spencer E. Dickson, A torsion theory for Abelian categories, Trans. Amer. Math. Soc. 121 (1966), 223–235. MR 191935, DOI 10.1090/S0002-9947-1966-0191935-0
Jonathan S. Golan, Localization of noncommutative rings, Pure and Applied Mathematics, No. 30, Marcel Dekker, Inc., New York, 1975. MR 0366961
Nathan Jacobson, Structure of rings, Revised edition, American Mathematical Society Colloquium Publications, Vol. 37, American Mathematical Society, Providence, R.I., 1964. MR 0222106
R. E. Johnson and E. T. Wong, Quasi-injective modules and irreducible rings, J. London Math. Soc. 36 (1961), 260–268. MR 131445, DOI 10.1112/jlms/s1-36.1.260
Joachim Lambek, Torsion theories, additive semantics, and rings of quotients, Lecture Notes in Mathematics, Vol. 177, Springer-Verlag, Berlin-New York, 1971. With an appendix by H. H. Storrer on torsion theories and dominant dimensions. MR 0284459

Lectures on rings and modules

Bo Stenström, Rings of quotients, Die Grundlehren der mathematischen Wissenschaften, Band 217, Springer-Verlag, New York-Heidelberg, 1975. An introduction to methods of ring theory. MR 0389953