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- W1574554306 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)." @default.
- W1574554306 created "2016-06-24" @default.
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- W1574554306 date "2006-01-01" @default.
- W1574554306 modified "2023-09-24" @default.
- W1574554306 title "Two statements about infinite products that are not quite true" @default.
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- W1574554306 doi "https://doi.org/10.1090/conm/420/07967" @default.
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