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Let $R$ be an associative ring with unit and let $N$ be a (unitary right $R$-) module (cf. also [[Associative rings and algebras|Associative rings and algebras]]; [[Module|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|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|Injective module]]). So, Whitehead test modules make it possible to test for projectivity (injectivity) of a module by computing a single <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120140/w12014012.png"/>-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|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|Set theory]]; [[Zermelo axiom|Zermelo axiom]]) that there are no p-test modules, [[#References|[a2]]], [[#References|[a8]]].

This is related to the structure of Whitehead modules ($M$ is a Whitehead module if $\operatorname { Ext } _ { R } ^ { 1 } ( M , R ) = 0$, [[#References|[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 [[#References|[a5]]], [[#References|[a6]]], and whose combinatorial equivalent was identified in [[#References|[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 [[#References|[a8]]].

Let $\kappa$ be a [[Cardinal number|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 [[#References|[a7]]]. There exist various non-Artinian $n$-saturated rings for $n &lt; \aleph_0$, but all $\aleph_{0}$-saturated rings are Artinian (cf. also [[Artinian ring|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|Morita equivalence]]) to the upper triangular $( 2 \times 2 )$-matrix ring over a skew-field [[#References|[a4]]] (cf. also [[Matrix ring|Matrix ring]]; [[Ring with division|Ring with division]]; [[Division algebra|Division algebra]])

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