Abstract
A module is called distributive (is said to be a chain module) if the lattice of all its submodules is distributive (is a chain). Let a ringA be a finitely generated module over its unitary central subringR. We prove the equivalence of the following conditions:
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(1)
A is a right or left distributive semiprime ring;
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(2)
for any maximal idealM of a subringR central inA, the ring of quotientsA M is a finite direct product of semihereditary Bézout domains whose quotient rings by the Jacobson radicals are finite direct products of skew fields;
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(3)
all right ideals and all left ideals of the ringA are flat (right and left) modules over the ringA, andA is a distributive ring, without nonzero nilpotent elements, all of whose quotient rings by prime ideals are semihereditary orders in skew fields.
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Translated fromMatematicheskie Zametki, Vol. 60, No. 2, pp. 254–277, August, 1996.
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Tuganbaev, A.A. Flat modules and rings finitely generated as modules over their center. Math Notes 60, 186–203 (1996). https://doi.org/10.1007/BF02305182
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DOI: https://doi.org/10.1007/BF02305182