Arxiv:1406.1932V2 [Math.GR] 2 Oct 2014 B Cited
Comments, corrections, and related references welcomed, as always! TEXed January 23, 2018 HOMOMORPHISMS ON INFINITE DIRECT PRODUCTS OF GROUPS, RINGS AND MONOIDS GEORGE M. BERGMAN Abstract. We study properties of a group, abelian group, ring, or monoid B which (a) guarantee that every homomorphism from an infinite direct product QI Ai of objects of the same sort onto B factors through the direct product of finitely many ultraproducts of the Ai (possibly after composition with the natural map B → B/Z(B) or some variant), and/or (b) guarantee that when a map does so factor (and the index set has reasonable cardinality), the ultrafilters involved must be principal. A number of open questions and topics for further investigation are noted. 1. Introduction A direct product Qi∈I Ai of infinitely many nontrivial algebraic structures is in general a “big” object: it has at least continuum cardinality, and if the operations of the Ai include a vector-space structure, it has at least continuum dimension. But there are many situations where the set of homomorphisms from such a product to a fixed object B is unexpectedly restricted. The poster child for this phenomenon is the case where the objects are abelian groups, and B is the infinite cyclic group. In that situation, if the index set I is countable (or, indeed, of less than an enormous cardinality – some details are recalled in §4), then every homomorphism Qi∈I Ai → B factors through the projection of Qi∈I Ai onto the product of finitely many of the Ai. An abelian group B which, like the infinite cyclic group, has this property, is called “slender”.
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