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The Anderson conjecture and a maximal class of monoids over which projective modules are free. (Russian) Zbl 0654.13013

The main result of this paper is a generalization of the well known problem of Serre about the freeness of the projective modules over polynomial rings \(K[X_ 1,...,X_ n]\), K a field, solved in 1976 by D. Quillen and A. A. Suslin. One shows: The maximal class of cancellative monoids, for which all finitely generated projective modules over monoidal R-algebras (R a domain with principal ideals) are free, coincides with the class of seminormal monoids without nontrivial torsion. An important step in the proof is the fact that normal monoids have the above property, as conjectured by D. F. Anderson [Pac. J. Math. 79, 5-17 (1978; Zbl 0372.13006)]. The proof is based on some combinatorial properties of monoids.
Reviewer: N.Manolache

MSC:

13C10 Projective and free modules and ideals in commutative rings
20M25 Semigroup rings, multiplicative semigroups of rings
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