We determine rings R with the property that all (finitely generated) nonsingular right R-modules have projective covers. These are just the rings with t-supplemented (finitely generated) free right modules. Hence, they are called right (finitely) Σ-t-supplemented. It is also shown that a ring R for which every cyclic nonsingular right R-module has a projective cover is exactly a right t-supplemented ring. It is proved that, for a continuous ring R, the property of right Σ- t-supplementedness is equivalent to the semisimplicity of R/Z 2(R R ), while the property of being right finitely Σ- t-supplemented is equivalent to the right self-injectivity of R/Z 2(R R ). Moreover, for a von Neumann regular ring R, the properties of being right Σ- t -supplemented, right finitely Σ- t -supplemented, and right t-supplemented are equivalent to the semisimplicity, right self-injectivity, and right continuity of R, respectively.
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Published in Ukrains’kyi Matematychnyi Zhurnal, Vol. 68, No. 1, pp. 3–13, January, 2016.
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Asgari, S., Haghany, A. Rings Whose Nonsingular Modules Have Projective Covers. Ukr Math J 68, 1–13 (2016). https://doi.org/10.1007/s11253-016-1204-7
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DOI: https://doi.org/10.1007/s11253-016-1204-7