Whitehead test module

Let be an associative ring with unit and let be a (unitary right -) module (cf. also Associative rings and algebras; Module). Then is a Whitehead test module for projectivity (or a p-test module) if for each module , implies is projective (cf. also Projective module). Dually, is a Whitehead test module for injectivity (or an i-test module) if for each module , implies is injective (cf. also Injective module). So, Whitehead test modules make it possible to test for projectivity (injectivity) of a module by computing a single -group.

By Baer's criterion, for any ring there is a proper class of i-test modules. Dually, for any right-perfect ring (cf. also Perfect ring) there is a proper class of p-test modules. If is not right perfect, then it is consistent with ZFC (cf. Set theory; Zermelo axiom) that there are no p-test modules, [a2], [a8].

This is related to the structure of Whitehead modules ( is a Whitehead module if , [a1]). If is a right-hereditary ring, then there is a cyclic p-test module if and only if is p-test if and only if every Whitehead module is projective. The validity of the latter for is the famous Whitehead problem, whose independence of ZFC was proved by S. Shelah [a5], [a6], and whose combinatorial equivalent was identified in [a3]. If is right hereditary but not right perfect, then it is consistent with ZFC that there is a proper class of p-test modules [a8].

Let be a cardinal number. Then is -saturated if each non-projective -generated module is an i-test module. If is -saturated for all cardinal numbers , then is called a fully saturated ring [a7]. There exist various non-Artinian -saturated rings for , but all -saturated rings are Artinian (cf. also Artinian ring). Moreover, all right-hereditary -saturated rings are fully saturated, and their class coincides with the class of rings , , or , where is completely reducible and is Morita equivalent (cf. Morita equivalence) to the upper triangular -matrix ring over a skew-field [a4] (cf. also Matrix ring; Ring with division; Division algebra)

References

[a1]	P.C. Eklof, A.H. Mekler, "Almost free modules: set-theoretic methods" , North-Holland (1990)
[a2]	P.C. Eklof, S. Shelah, "On Whitehead modules" J. Algebra , 142 (1991) pp. 492–510
[a3]	P.C. Eklof, S. Shelah, "A combinatorial principle equivalent to the existence of non-free Whitehead groups" Contemp. Math. , 171 (1994) pp. 79–98
[a4]	P.C. Eklof, J. Trlifaj, "How to make Ext vanish" preprint
[a5]	S. Shelah, "Infinite abelian groups, Whitehead problem and some constructions" Israel J. Math. , 18 (1974) pp. 243–256
[a6]	S. Shelah, "A compactness theorem for singular cardinals, free algebras, Whitehead problem and transversals" Israel J. Math. , 21 (1975) pp. 319–349
[a7]	J. Trlifaj, "Associative rings and the Whitehead property of modules" , Algebra Berichte , 63 , R. Fischer (1990)
[a8]	J. Trlifaj, "Whitehead test modules" Trans. Amer. Math. Soc. , 348 (1996) pp. 1521–1554