Namespaces
Variants
Actions

Difference between revisions of "Whitehead test module"

From Encyclopedia of Mathematics
Jump to: navigation, search
(Importing text file)
 
m (AUTOMATIC EDIT (latexlist): Replaced 37 formulas out of 38 by TEX code with an average confidence of 2.0 and a minimal confidence of 2.0.)
Line 1: Line 1:
Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120140/w1201401.png" /> be an associative ring with unit and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120140/w1201402.png" /> be a (unitary right <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120140/w1201403.png" />-) module (cf. also [[Associative rings and algebras|Associative rings and algebras]]; [[Module|Module]]). Then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120140/w1201404.png" /> is a Whitehead test module for projectivity (or a p-test module) if for each module <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120140/w1201405.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120140/w1201406.png" /> implies <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120140/w1201407.png" /> is projective (cf. also [[Projective module|Projective module]]). Dually, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120140/w1201408.png" /> is a Whitehead test module for injectivity (or an i-test module) if for each module <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120140/w1201409.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120140/w12014010.png" /> implies <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120140/w12014011.png" /> 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.
+
<!--This article has been texified automatically. Since there was no Nroff source code for this article,
 +
the semi-automatic procedure described at https://encyclopediaofmath.org/wiki/User:Maximilian_Janisch/latexlist
 +
was used.
 +
If the TeX and formula formatting is correct, please remove this message and the {{TEX|semi-auto}} category.
  
By Baer's criterion, for any ring <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120140/w12014013.png" /> 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120140/w12014014.png" /> 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]]].
+
Out of 38 formulas, 37 were replaced by TEX code.-->
  
This is related to the structure of Whitehead modules (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120140/w12014015.png" /> is a Whitehead module if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120140/w12014016.png" />, [[#References|[a1]]]). If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120140/w12014017.png" /> is a right-hereditary ring, then there is a cyclic p-test module if and only if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120140/w12014018.png" /> is p-test if and only if every Whitehead module is projective. The validity of the latter for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120140/w12014019.png" /> 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120140/w12014020.png" /> 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]]].
+
{{TEX|semi-auto}}{{TEX|partial}}
 +
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.
  
Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120140/w12014021.png" /> be a [[Cardinal number|cardinal number]]. Then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120140/w12014022.png" /> is <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120140/w12014024.png" />-saturated if each non-projective <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120140/w12014025.png" />-generated module is an i-test module. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120140/w12014026.png" /> is <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120140/w12014027.png" />-saturated for all cardinal numbers <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120140/w12014028.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120140/w12014029.png" /> is called a fully saturated ring [[#References|[a7]]]. There exist various non-Artinian <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120140/w12014030.png" />-saturated rings for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120140/w12014031.png" />, but all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120140/w12014032.png" />-saturated rings are Artinian (cf. also [[Artinian ring|Artinian ring]]). Moreover, all right-hereditary <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120140/w12014033.png" />-saturated rings are fully saturated, and their class coincides with the class of rings <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120140/w12014034.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120140/w12014035.png" />, or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120140/w12014036.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120140/w12014037.png" /> is completely reducible and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120140/w12014038.png" /> is Morita equivalent (cf. [[Morita equivalence|Morita equivalence]]) to the upper triangular <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120140/w12014039.png" />-matrix ring over a skew-field [[#References|[a4]]] (cf. also [[Matrix ring|Matrix ring]]; [[Ring with division|Ring with division]]; [[Division algebra|Division algebra]])
+
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====
 
====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>
+
<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>

Revision as of 16:55, 1 July 2020

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
How to Cite This Entry:
Whitehead test module. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Whitehead_test_module&oldid=16198
This article was adapted from an original article by Jan Trlifaj (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article