Whitehead test module

Let $R$ be an associative ring with unit and let $N$ be a (unitary right $R$-) module (cf. also Associative rings and algebras; Module). Then $N$ is a Whitehead test module for projectivity (or a p-test module) if for each module $M$, $\operatorname { Ext } _ { R } ^ { 1 } ( M , N ) = 0$ implies $M$ is projective (cf. also Projective module). Dually, $N$ is a Whitehead test module for injectivity (or an i-test module) if for each module $M$, $\operatorname { Ext } _ { R } ^ { 1 } ( N , M ) = 0$ implies $M$ 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 $R$ 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 $R$ 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 ($M$ is a Whitehead module if $\operatorname { Ext } _ { R } ^ { 1 } ( M , R ) = 0$, [a1]). If $R$ is a right-hereditary ring, then there is a cyclic p-test module if and only if $R$ is p-test if and only if every Whitehead module is projective. The validity of the latter for $R = \mathbf{Z}$ 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 $R$ 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 $\kappa$ be a cardinal number. Then $R$ is $\kappa$-saturated if each non-projective $\leq \kappa$-generated module is an i-test module. If $R$ is $\lambda$-saturated for all cardinal numbers $\lambda$, then $R$ is called a fully saturated ring [a7]. There exist various non-Artinian $n$-saturated rings for $n < \aleph_0$, but all $\aleph_{0}$-saturated rings are Artinian (cf. also Artinian ring). Moreover, all right-hereditary $\aleph_{0}$-saturated rings are fully saturated, and their class coincides with the class of rings $S$, $T$, or $S \boxplus T$, where $S$ is completely reducible and $T$ is Morita equivalent (cf. Morita equivalence) to the upper triangular $( 2 \times 2 )$-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