Big projective modules are free

IntroductionFinitely generated proiective modules rise significantly in certain geo- metric nd rithmetic questions.We shll show here that nonfinitely generated proiective modules, in contrast, invite little interest; for we show that n obviously necessary "connectedness" condition for such module to be free is lso sufficient.More precisely, cll n R-module P uniformly -big, where b is n infinite crdinl, if (i) P cn be generated by b elements, nd (ii) PlOP requires generators for ll two-sided ideals R).A free module with bsis of elements is mnifestly uniformly -big.Our min result (Corollary 3.2) sserts, conversely, that, with suitable chain conditions on R, a uniformly big projective R-module is free.Finally, we sk, for wht R re 11 nonfinitely generated proiective modules uniformly big?For commutative rings, the nswer is quite stisfctory; with mild ssumptions one requires only that spec (R) be connected (i.e., that there exist no nontrivil idempotents).For R Zr, with r finite group, Swan (unpublished) hs established this conclusion when r is solvable, ad it is undoubtedly true in general.Our method relies on two bsic tools.One, nturlly enough, is Kplnsky's remarkable theorem [2, Theorem 1] which sserts that every pro- ]ective module is direct sum of countbly generated modules.The second is n elegant little swindle, observed several years go by Eilenberg, nd which might well hve sprung from the brow of Brry M:zur.It is this result, recorded below, which permits us to wive the delicate rithmetic questions which plague the finitely generated cse.EILENBERG'S LEMMA.If P @ Q F with F a nonfinitely generated free module, then P @ F _N F.Proof.