Two Statements about Infinite Products that Are Not Quite True

The first half of this note concerns modules; so let R be a nonzero associative ring with unit. A countably infinite direct product of copies of R, which we will usually regard as a left R-module, will be written R (ω denoting the set of natural numbers); the corresponding direct sum, i.e., the free left R-module of countably infinite rank, will be written ⊕ ω R. Here, now, are the two not-always-true statements of the title: (1) There is no surjective left R-module homomorphism ⊕ ω R→ R. (2) There is no surjective left R-module homomorphism R → ⊕ ω R. In §2 we will note classes of rings R for which each of these statements fails. In §§3-4, however, we will see that (1) holds, i.e., R requires uncountably many generators as a left R-module, unless R is finitely generated, and that (2) holds unless R has descending chain condition on finitely generated right ideals. From the above assertion regarding (1), and the statement of (2), it is easy to see that for every R, at least one of (1), (2) holds. We shall also see that the above restriction on rings for which (2) fails implies that for every R, either (2) or the statement (3) There is no embedding of left R-modules R → ⊕ ω R. holds. (I did not count (3) among the “not quite true” statements of the title, because it does not appear that it is “nearly” true; i.e., that its failure implies strong restrictions on R.) The result asserted above in connection with (1) will in fact be proved with R replaced by M for M any R-module, while the result on (2) will be obtained with R generalized to the inverse limit of any countable inverse system of finitely generated R-modules and surjective homomorphisms.