We continue the study of modules over a general ring R whose submodules have a unique closure relative to a hereditary torsion theory on Mod-R. It is proved that, for a given ring R and a hereditary torsion theory τ on Mod-R, every submodule of every right R-module has a unique closure with respect to τ if and only if τ is generated by projective simple right R-modules. In particular, a ring R is a right Kasch ring if and only if every submodule of every right R-module has a unique closure with respect to the Lambek torsion theory.
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Published in Ukrains’kyi Matematychnyi Zhurnal, Vol. 66, No. 7, pp. 922–929, July, 2014.
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Doğruöz, S., Harmanci, A. & Smith, P.F. Modules with Unique Closure Relative to a Torsion Theory. III. Ukr Math J 66, 1028–1036 (2014). https://doi.org/10.1007/s11253-014-0992-x
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DOI: https://doi.org/10.1007/s11253-014-0992-x