SherbyNet - /page/wikibook/entry.mu

Micron Document

<!DOCTYPE html>

<title>Algebraically compact module</title>

</head>

Algebraically compact module

</h1>

</header>

In <a href="Mathematics" title="Mathematics">mathematics</a>, algebraically compact modules, also called pure-injective modules, are <a href="Module_(mathematics)" title="Module (mathematics)">modules</a> that have a certain "nice" property which allows the solution of infinite systems of equations in the module by <a href="Finitary" title="Finitary">finitary</a> means. The solutions to these systems allow the extension of certain kinds of <a href="Module_homomorphism" title="Module homomorphism">module homomorphisms</a>. These algebraically compact modules are analogous to <a href="Injective_module" title="Injective module">injective modules</a>, where one can extend all module homomorphisms. All injective modules are algebraically compact, and the analogy between the two is made quite precise by a category embedding.

<div class="mw-heading mw-heading2"><h2 id="Definitions">Definitions</h2></div>

Let R be a <a href="Ring_(mathematics)" title="Ring (mathematics)">ring</a>, and M a left R-module. Consider a system of infinitely many linear equations

</mrow>

</munder>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \sum _{j\in J}r_{i,j}x_{j}=m_{i},}</annotation>

</semantics>

</math><img src="./d8b6a14cb6aac98443625aca24920ac93f10b37e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.338ex; width:15.55ex; height:5.843ex;" alt="{\displaystyle \sum _{j\in J}r_{i,j}x_{j}=m_{i},}" loading="lazy"></dd></dl>

where both sets I and J may be infinite, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle m_{i}\in M,}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle m_{i}\in M,}</annotation>

</semantics>

</math><img src="./a60c0d7a7660c997a651fec63775dba7f66552ce.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:8.77ex; height:2.509ex;" alt="{\displaystyle m_{i}\in M,}" loading="lazy"> and for each i the number of nonzero <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r_{i,j}\in R}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle r_{i,j}\in R}</annotation>

</semantics>

</math><img src="./2101c71d786b5eee5c514c385b966fec66ae5ed1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:7.588ex; height:2.843ex;" alt="{\displaystyle r_{i,j}\in R}" loading="lazy"> is finite.

The goal is to decide whether such a system has a solution, that is whether there exist elements xj of M such that all the equations of the system are simultaneously satisfied. (It is not required that only finitely many xj are non-zero.)

The module M is algebraically compact if, for all such systems, if every subsystem formed by a finite number of the equations has a solution, then the whole system has a solution. (The solutions to the various subsystems may be different.)

On the other hand, a <a href="Module_homomorphism" title="Module homomorphism">module homomorphism</a> M → K is a pure embedding if the <a href="Induced_homomorphism" title="Induced homomorphism">induced homomorphism</a> between the <a href="Tensor_product" title="Tensor product">tensor products</a> C ⊗ M → C ⊗ K is <a href="Injective" class="mw-redirect" title="Injective">injective</a> for every right R-module C. The module M is pure-injective if any pure injective homomorphism j : M → K <a href="Split_short_exact_sequence" class="mw-redirect" title="Split short exact sequence">splits</a> (that is, there exists f : K → M with <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f\circ j=1_{M}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f\circ j=1_{M}}</annotation>

</semantics>

</math><img src="./97cddb843c0373b64a8d97dd7567e8cfb9b9b16f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.651ex; height:2.509ex;" alt="{\displaystyle f\circ j=1_{M}}" loading="lazy">).

It turns out that a module is algebraically compact <a href="If_and_only_if" title="If and only if">if and only if</a> it is pure-injective.

<div class="mw-heading mw-heading2"><h2 id="Examples">Examples</h2></div>

All modules with finitely many elements are algebraically compact.

Every <a href="Vector_space" title="Vector space">vector space</a> is algebraically compact (since it is pure-injective). More generally, every <a href="Injective_module" title="Injective module">injective module</a> is algebraically compact, for the same reason.

If R is an <a href="Associative_algebra" title="Associative algebra">associative algebra</a> with 1 over some <a href="Field_(mathematics)" title="Field (mathematics)">field</a> k, then every R-module with finite k-<a href="Dimension_of_a_vector_space" class="mw-redirect" title="Dimension of a vector space">dimension</a> is algebraically compact. This, together with the fact that all finite modules are algebraically compact, gives rise to the intuition that algebraically compact modules are those (possibly "large") modules which share the nice properties of "small" modules.

The <a href="Pr%C3%BCfer_group" title="Prüfer group">Prüfer groups</a> are algebraically compact <a href="Abelian_group" title="Abelian group">abelian groups</a> (i.e. Z-modules). The ring of <a href="P-adic_number" title="P-adic number">p-adic integers</a> for each prime p is algebraically compact as both a module over itself and a module over Z. The <a href="Rational_number" title="Rational number">rational numbers</a> are algebraically compact as a Z-module. Together with the <a href="Indecomposable_module" title="Indecomposable module">indecomposable</a> finite modules over Z, this is a complete list of indecomposable algebraically compact modules.

Many algebraically compact modules can be produced using the <a href="Injective_cogenerator" title="Injective cogenerator">injective cogenerator</a> Q/Z of abelian groups. If H is a right module over the ring R, one forms the (algebraic) character module H* consisting of all <a href="Group_homomorphism" title="Group homomorphism">group homomorphisms</a> from H to Q/Z. This is then a left R-module, and the *-operation yields a <a href="Faithful_functor" class="mw-redirect" title="Faithful functor">faithful</a> contravariant <a href="Functor" title="Functor">functor</a> from right R-modules to left R-modules.

Every module of the form H* is algebraically compact. Furthermore, there are pure injective homomorphisms H → H**, <a href="Natural_transformation" title="Natural transformation">natural</a> in H. One can often simplify a problem by first applying the *-functor, since algebraically compact modules are easier to deal with.

<div class="mw-heading mw-heading2"><h2 id="Facts">Facts</h2></div>

The following condition is equivalent to M being algebraically compact:

<ul><li>For every index set I, the addition map M(I) → M can be extended to a module homomorphism MI → M (here M(I) denotes the <a href="Direct_sum_of_modules" title="Direct sum of modules">direct sum</a> of copies of M, one for each element of I; MI denotes the <a href="Product_(category_theory)" title="Product (category theory)">product</a> of copies of M, one for each element of I).</li></ul>

Every <a href="Indecomposable_module" title="Indecomposable module">indecomposable</a> algebraically compact module has a <a href="Local_ring" title="Local ring">local</a> <a href="Endomorphism_ring" title="Endomorphism ring">endomorphism ring</a>.

Algebraically compact modules share many other properties with injective objects because of the following: there exists an embedding of R-Mod into a <a href="Grothendieck_category" title="Grothendieck category">Grothendieck category</a> G under which the algebraically compact R-modules precisely correspond to the injective objects in G.

Every R-module is <a href="Elementary_equivalence" title="Elementary equivalence">elementary equivalent</a> to an algebraically compact R-module and to a direct sum of <a href="Indecomposable_module" title="Indecomposable module">indecomposable</a> algebraically compact R-modules.<a href="#cite_note-1">[1]</a>

<div class="mw-heading mw-heading2"><h2 id="References">References</h2></div>

/* start https://en.wikipedia.org/ */

.mw-parser-output .reflist{margin-bottom:0.5em;list-style-type:decimal}@media screen{.mw-parser-output .reflist{font-size:90%}}.mw-parser-output .reflist .references{font-size:100%;margin-bottom:0;list-style-type:inherit}.mw-parser-output .reflist-columns-2{column-width:30em}.mw-parser-output .reflist-columns-3{column-width:25em}.mw-parser-output .reflist-columns{margin-top:0.3em}.mw-parser-output .reflist-columns ol{margin-top:0}.mw-parser-output .reflist-columns li{page-break-inside:avoid;break-inside:avoid-column}.mw-parser-output .reflist-upper-alpha{list-style-type:upper-alpha}.mw-parser-output .reflist-upper-roman{list-style-type:upper-roman}.mw-parser-output .reflist-lower-alpha{list-style-type:lower-alpha}.mw-parser-output .reflist-lower-greek{list-style-type:lower-greek}.mw-parser-output .reflist-lower-roman{list-style-type:lower-roman}

/* end https://en.wikipedia.org/ */

</style><div class="reflist">

/* start https://en.wikipedia.org/ */

.mw-parser-output cite.citation{font-style:inherit;word-wrap:break-word}.mw-parser-output .citation q{quotes:"\"""\"""'""'"}.mw-parser-output .citation:target{background-color:rgba(0,127,255,0.133)}.mw-parser-output .id-lock-free.id-lock-free a{background:url("./mw/Lock-green.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-limited.id-lock-limited a,.mw-parser-output .id-lock-registration.id-lock-registration a{background:url("./mw/Lock-gray-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-subscription.id-lock-subscription a{background:url("./mw/Lock-red-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .cs1-ws-icon a{background:url("./mw/Wikisource-logo.svg")right 0.1em center/12px no-repeat}body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-free a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-limited a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-registration a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-subscription a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .cs1-ws-icon a{background-size:contain;padding:0 1em 0 0}.mw-parser-output .cs1-code{color:inherit;background:inherit;border:none;padding:inherit}.mw-parser-output .cs1-hidden-error{display:none;color:var(--color-error,#d33)}.mw-parser-output .cs1-visible-error{color:var(--color-error,#d33)}.mw-parser-output .cs1-maint{display:none;color:#085;margin-left:0.3em}.mw-parser-output .cs1-kern-left{padding-left:0.2em}.mw-parser-output .cs1-kern-right{padding-right:0.2em}.mw-parser-output .citation .mw-selflink{font-weight:inherit}@media screen{.mw-parser-output .cs1-format{font-size:95%}html.skin-theme-clientpref-night .mw-parser-output .cs1-maint{color:#18911f}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .cs1-maint{color:#18911f}}

/* end https://en.wikipedia.org/ */

</style><cite id="CITEREFPrest1988" class="citation book cs1">Prest, Mike (1988). Model theory and modules. London Mathematical Society Lecture Note Series: Cambridge University Press, Cambridge. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>0-521-34833-1</bdi>.</cite>

</li>

</ol></div></div>

<ul><li>C.U. Jensen and H. Lenzing: Model Theoretic Algebra, Gordon and Breach, 1989</li></ul></div><div><div class="zim-footer">

This article is issued from <a class="external text" title="Last edited on 2025-06-07" href="https://en.wikipedia.org/wiki/?title=Algebraically_compact_module&oldid=1294472448">Wikipedia</a>. The text is available under <a class="external text" href="https://creativecommons.org/licenses/by-sa/4.0/deed.en">Creative Commons Attribution-Share Alike 4.0</a> unless otherwise noted. Additional terms may apply for the media files.

</div>

</div></div>

</div>

</main>

</div>

</body></html>