Two statements about infinite products that are not quite true

Abstract

Hard to summarize concisely; here are the high points. The first two statements below are ring-theoretic; in these R is a nontrivial ring, R^\omega, and \bigoplus_\omega R are the direct product, respectively direct sum, of countably many copies of R; the remaining two statements are in the context of general algebra (a.k.a. universal algebra): (i) There exist nontrivial rings R for which one has surjective homomorphisms \bigoplus_\omega R -> R^\omega -- but in such cases, R^\omega is in fact finitely generated as a left R-module. (ii) There exist nontrivial rings R for which one has surjective homomorphisms R^\omega -> \bigoplus_\omega R -- but in such cases, R must have DCC on finitely generated right ideals. (iii) The full permutation group S on an infinite set \Omega has the property that the |\Omega|-fold direct product of copies of S is generated over its diagonal subgroup by a single element. (iv) Whenever an algebra S in the sense of universal algebra has the property that the countable direct product S^\omega is finitely generated over its diagonal subalgebra (or even when the corresponding property holds with an ultrapower in place of this direct product), S has some of the other strange properties known to hold for infinite symmetric groups (cf. math.GR/0401304).