Equalizers and Flatness Properties of Acts,II |
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Authors: | Email author" target="_blank">S?Bulman-FlemingEmail author Email author" target="_blank">M?KilpEmail author |
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Institution: | (1) Department of Mathematics, Wilfrid Laurier University, Waterloo, ON N2L 3C5, Canada;(2) Institute of Pure Mathematics, University of Tartu, 50090 Tartu, Estonia |
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Abstract: | In Comm. Algebra 30 (3) (2002), 1475–1498, Bulman-Fleming and Kilp
developed various notions of flatness of a right act AS over a monoid S that are based on the extent to which the functor AS$\otimes -$
preserves equalizers. In Semigroup Forum 65 (3) (2002), 428–449,
Bulman-Fleming discussed in detail one of these notions,
annihilator-flatness. The present paper is devoted to two more of these
notions, weak equalizer-flatness and strong torsion-freeness. An act AS
is called weakly equalizer-flat if the functor AS$\otimes -$ almost
preserves equalizers of any two homomorphisms into the left act SS, and
strongly torsion-free if this functor almost preserves equalizers of any
two homomorphisms from SS into the Rees factor act S(S/Sc), where c is any right cancellable element of S. (The adverb almost signifies
that the canonical morphism provided by the universal property of equalizers
may be only a monomorphism rather than an isomorphism.) From the definitions
it is clear that flatness implies weak equalizer-flatness, which in turn
implies annihilator-flatness, and it was known already that both of these
implications are strict. A monoid is called right absolutely weakly
equalizer-flat if all of its right acts are weakly equalizer-flat. In this
paper we prove a result which implies that right PP monoids with central
idempotents are absolutely weakly equalizer-flat. We also show that for a
relatively large class of commutative monoids, right absolute
equalizer-flatness and right absolute annihilator-flatness coincide.
Finally, we provide examples showing that the implication between strong
torsion-freeness and torsion-freeness is strict. |
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