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| In [[mathematics]], in the field of [[abstract algebra]], the '''structure theorem for finitely generated modules over a principal ideal domain''' is a generalization of the [[fundamental theorem of finitely generated abelian groups]] and roughly states that finitely generated modules can be uniquely decomposed in much the same way that integers have a [[prime factorization]]. The result provides a simple framework to understand various canonical form results for square matrices over fields.
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| ==Statement==
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| When a vector space over a [[Field (mathematics)|field]] ''F'' has a finite generating set, then one may extract from it a [[basis (vector space)|basis]] consisting of a finite number ''n'' of vectors, and the space is therefore isomorphic to ''F<sup>n</sup>''. The corresponding statement with the ''F'' generalized to a [[principal ideal domain]] ''R'' is no longer true, as a [[finitely generated module]] over ''R'' need not have any basis. However such a module is still isomorphic to a quotient of some module ''R<sup>n</sup>'' with ''n'' finite (to see this it suffices to construct the morphism that sends the elements of the canonical basis ''R<sup>n</sup>'' to the generators of the module, and take the quotient by its [[kernel (algebra)|kernel]].) By changing the choice of generating set, one can in fact describe the module as the quotient of some ''R<sup>n</sup>'' by a particularly simple submodule, and this is the structure theorem.
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| The structure theorem for [[finitely generated module]]s over a [[principal ideal domain]] usually appears in the following two forms.
| | '''MathML''' |
| | :<math forcemathmode="mathml">E=mc^2</math> |
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| ===Invariant factor decomposition===
| | <!--'''PNG''' (currently default in production) |
| For every finitely generated module {{math|''M''}} over a principal ideal domain {{math|''R''}}, there is a unique decreasing sequence of proper ideals <math>(d_1)\supseteq(d_2)\supseteq\cdots\supseteq(d_n)</math> such that {{math|''M''}} isomorphic to the sum of cyclic modules:
| | :<math forcemathmode="png">E=mc^2</math> |
| :<math>M\cong\bigoplus_i R/(d_i) = R/(d_1)\oplus R/(d_2)\oplus\cdots\oplus R/(d_n).</math> | |
| The generators <math>d_i</math> of the ideals are unique up to multiplication by a [[unit (ring theory)|unit]], and are called [[invariant factor]]s of ''M''. Since the ideals should be proper, these factors must not themselves be invertible (this avoids trivial factors in the sum), and the inclusion of the ideals means one has divisibility <math>d_1\mid d_2\mid\cdots\mid d_n</math>. The free part is visible in the part of the decomposition corresponding to factors <math>d_i = 0</math>. Such factors, if any, occur at the end of the sequence.
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| While the direct sum is uniquely determined by {{math|''M''}}, the isomorphism giving the decomposition itself is ''not unique'' in general. For instance if {{math|''R''}} is actually a field, then all occurring ideals must be zero, and one obtains the decomposition of a finite dimensional vector space into a direct sum of one-dimensional subspaces; the number of such factors is fixed, namely the dimension of the space, but there is in a lot of freedom for choosing the subspaces themselves (if {{math|dim ''M'' > 1}}).
| | '''source''' |
| | :<math forcemathmode="source">E=mc^2</math> --> |
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| The nonzero <math>d_i</math> elements, together with the number of <math>d_i</math> which are zero, form a [[complete set of invariants]] for the module. Explicitly, this means that any two modules sharing the same set of invariants are necessarily isomorphic.
| | <span style="color: red">Follow this [https://en.wikipedia.org/wiki/Special:Preferences#mw-prefsection-rendering link] to change your Math rendering settings.</span> You can also add a [https://en.wikipedia.org/wiki/Special:Preferences#mw-prefsection-rendering-skin Custom CSS] to force the MathML/SVG rendering or select different font families. See [https://www.mediawiki.org/wiki/Extension:Math#CSS_for_the_MathML_with_SVG_fallback_mode these examples]. |
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| Some prefer to write the free part of ''M'' separately:
| | ==Demos== |
| :<math>R^f \oplus \bigoplus_i R/(d_i) = R^f \oplus R/(d_1)\oplus R/(d_2)\oplus\cdots\oplus R/(d_{n-f})</math>
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| where the visible <math>d_i</math> are nonzero, and ''f'' is the number of <math>d_i</math>'s in the original sequence which are 0.
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| ===Primary decomposition===
| | Here are some [https://commons.wikimedia.org/w/index.php?title=Special:ListFiles/Frederic.wang demos]: |
| :Every [[finitely generated module]] ''M'' over a [[principal ideal domain]] ''R'' is isomorphic to one of the form
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| ::<math>\bigoplus_i R/(q_i)</math> | |
| :where <math>(q_i) \neq R</math> and the <math>(q_i)</math> are [[primary ideal]]s. The <math>q_i</math> are unique (up to multiplication by units).
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| The elements <math>q_i</math> are called the ''elementary divisors'' of ''M''. In a PID, nonzero primary ideals are powers of primes, and so <math>(q_i)=(p_i^{r_i}) = (p_i)^{r_i}</math>. When <math>q_i=0</math>, the resulting indecomposable module is <math>R</math> itself, and this is inside the part of ''M'' that is a free module.
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| The summands <math>R/(q_i)</math> are [[indecomposable module|indecomposable]], so the primary decomposition is a decomposition into indecomposable modules, and thus every finitely generated module over a PID is a [[indecomposable module|completely decomposable module]]. Since PID's are Noetherian rings, this can be seen as a manifestation of the [[Lasker-Noether theorem]].
| | * accessibility: |
| | ** Safari + VoiceOver: [https://commons.wikimedia.org/wiki/File:VoiceOver-Mac-Safari.ogv video only], [[File:Voiceover-mathml-example-1.wav|thumb|Voiceover-mathml-example-1]], [[File:Voiceover-mathml-example-2.wav|thumb|Voiceover-mathml-example-2]], [[File:Voiceover-mathml-example-3.wav|thumb|Voiceover-mathml-example-3]], [[File:Voiceover-mathml-example-4.wav|thumb|Voiceover-mathml-example-4]], [[File:Voiceover-mathml-example-5.wav|thumb|Voiceover-mathml-example-5]], [[File:Voiceover-mathml-example-6.wav|thumb|Voiceover-mathml-example-6]], [[File:Voiceover-mathml-example-7.wav|thumb|Voiceover-mathml-example-7]] |
| | ** [https://commons.wikimedia.org/wiki/File:MathPlayer-Audio-Windows7-InternetExplorer.ogg Internet Explorer + MathPlayer (audio)] |
| | ** [https://commons.wikimedia.org/wiki/File:MathPlayer-SynchronizedHighlighting-WIndows7-InternetExplorer.png Internet Explorer + MathPlayer (synchronized highlighting)] |
| | ** [https://commons.wikimedia.org/wiki/File:MathPlayer-Braille-Windows7-InternetExplorer.png Internet Explorer + MathPlayer (braille)] |
| | ** NVDA+MathPlayer: [[File:Nvda-mathml-example-1.wav|thumb|Nvda-mathml-example-1]], [[File:Nvda-mathml-example-2.wav|thumb|Nvda-mathml-example-2]], [[File:Nvda-mathml-example-3.wav|thumb|Nvda-mathml-example-3]], [[File:Nvda-mathml-example-4.wav|thumb|Nvda-mathml-example-4]], [[File:Nvda-mathml-example-5.wav|thumb|Nvda-mathml-example-5]], [[File:Nvda-mathml-example-6.wav|thumb|Nvda-mathml-example-6]], [[File:Nvda-mathml-example-7.wav|thumb|Nvda-mathml-example-7]]. |
| | ** Orca: There is ongoing work, but no support at all at the moment [[File:Orca-mathml-example-1.wav|thumb|Orca-mathml-example-1]], [[File:Orca-mathml-example-2.wav|thumb|Orca-mathml-example-2]], [[File:Orca-mathml-example-3.wav|thumb|Orca-mathml-example-3]], [[File:Orca-mathml-example-4.wav|thumb|Orca-mathml-example-4]], [[File:Orca-mathml-example-5.wav|thumb|Orca-mathml-example-5]], [[File:Orca-mathml-example-6.wav|thumb|Orca-mathml-example-6]], [[File:Orca-mathml-example-7.wav|thumb|Orca-mathml-example-7]]. |
| | ** From our testing, ChromeVox and JAWS are not able to read the formulas generated by the MathML mode. |
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| As before, it is possible to write the free part (where <math>q_i=0</math>) separately and express ''M'' as:
| | ==Test pages == |
| :<math>R^f \oplus(\bigoplus_i R/(q_i))</math>
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| where the visible <math>q_i </math> are nonzero.
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| ==Proofs==
| | To test the '''MathML''', '''PNG''', and '''source''' rendering modes, please go to one of the following test pages: |
| One proof proceeds as follows:
| | *[[Displaystyle]] |
| * Every [[finitely generated module]] over a PID is also [[finitely presented module|finitely presented]] because a PID is [[noetherian ring|Noetherian]], an even stronger condition than [[coherent ring|coherence]]. | | *[[MathAxisAlignment]] |
| * Take a presentation, which is a map <math>R^r \to R^g</math> (relations to generators), and put it in [[Smith normal form]]. | | *[[Styling]] |
| This yields the invariant factor decomposition, and the diagonal entries of Smith normal form are the invariant factors.
| | *[[Linebreaking]] |
| | *[[Unique Ids]] |
| | *[[Help:Formula]] |
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| Another outline of a proof:
| | *[[Inputtypes|Inputtypes (private Wikis only)]] |
| * Denote by ''tM'' the [[torsion submodule]] of M. Then ''M''/''tM'' is a finitely generated [[torsion-free module|torsion free]] module, and such a module over a commutative PID is a [[free module]] of finite rank, so it is isomorphic to <math>R^n</math> for a positive integer ''n''. This free module can be embedded as a submodule ''F'' of ''M'', such that the embedding splits (is a right inverse of) the projection map; it suffices to lift each of the generators of ''F'' into ''M''. As a consequence <math>M= tM\oplus F</math>. | | *[[Url2Image|Url2Image (private Wikis only)]] |
| * For a prime ''p'' in ''R'' we can then speak of <math>N_p= \{m\in tM\mid \exists i, mp^i=0\}</math> for each prime ''p''. This is a submodule of ''tM'', and it turns out that each ''N''<sub>''p''</sub> is a direct sum of cyclic modules, and that ''tM'' is a direct sum of ''N''<sub>''p''</sub> for a finite number of distinct primes ''p''.
| | ==Bug reporting== |
| * Putting the previous two steps together, ''M'' is decomposed into cyclic modules of the indicated types.
| | If you find any bugs, please report them at [https://bugzilla.wikimedia.org/enter_bug.cgi?product=MediaWiki%20extensions&component=Math&version=master&short_desc=Math-preview%20rendering%20problem Bugzilla], or write an email to math_bugs (at) ckurs (dot) de . |
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| <!-- (commented out because it is a bit misleading and contains errors) * A finitely generated module is [[projection (mathematics)|projective]] if and only if it is [[localization of a module|locally]] [[free module|free]].
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| * PIDs are [[Dedekind domains]], i.e., they are [[Noetherian ring|Noetherian]] and their [[Localization of a ring|localizations]] are [[discrete valuation ring]]s (DVRs). | |
| * Torsion free modules over DVRs are free. Hence, torsion free modules over PIDs are projective.
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| * ''M/tM'' is torsion free, hence projective. Thus, ''M'' can be written as a direct sum of its torsion part and a projective part (in fact a free part), though not uniquely.
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| *:That is, there is always a [[short exact sequence]] <math>0 \to tM \to M \to M/tM \to 0,</math> as the torsion part of a module is a submodule. By projectivity of <math>M/tM,</math> this has a splitting (a map <math>M/tM \to M</math> such that <math>M/tM \to M \to M/tM</math> is the identity).
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| * If ''M'' is projective, so is a direct summand of a free module ''F = M + N''. One proves that ''N'' is locally 0, and hence is 0. Therefore, ''M'' is free.
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| * If ''M'' is torsion, it is the quotient of a free module. Using the ideas of the previous part, one proves it is a quotient by a free submodule, which must have rank equal to the original module. That is, torsion modules are finitely presented. Now use [[Smith normal form]].-->
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| ==Corollaries== | |
| This includes the classification of [[finite-dimensional vector space]]s as a special case, where <math>R = K</math>. Since fields have no non-trivial ideals, every finitely generated vector space is free.
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| Taking <math>R=\mathbb{Z}</math> yields the [[fundamental theorem of finitely generated abelian groups]].
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| Let ''T'' be a linear operator on a [[finite-dimensional vector space]] ''V'' over ''K''. Taking <math>R=K[T]</math>, the algebra of polynomials with coefficients in ''K'' evaluated at ''T'', yields structure information about ''T''. ''V'' can be viewed as a finitely generated module over <math>K[T]</math>. The last invariant factor is the [[Minimal polynomial (field theory)|minimal polynomial]], and the product of invariant factors is the [[characteristic polynomial]]. Combined with a standard matrix form for <math>K[T]/p(T)</math>, this yields various [[canonical form]]s:
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| * [[invariant factors]] + [[companion matrix]] yields [[Frobenius normal form]] (aka, [[rational canonical form]])
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| * [[primary decomposition]] + [[companion matrix]] yields [[primary rational canonical form]]
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| * [[primary decomposition]] + [[Jordan block]]s yields [[Jordan canonical form]] (this latter only holds over an [[algebraically closed field]])
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| ==Uniqueness== | |
| While the invariants (rank, invariant factors, and elementary divisors) are unique, the isomorphism between ''M'' and its [[canonical form]] is not unique, and does not even preserve the [[direct sum of modules|direct sum]] decomposition. This follows because there are non-trivial automorphisms of these modules which do not preserve the summands.
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| However, one has a canonical torsion submodule ''T'', and similar canonical submodules corresponding to each (distinct) invariant factor, which yield a canonical sequence:
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| :<math>0 < \cdots < T < M.</math>
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| Compare [[composition series]] in [[Jordan–Hölder theorem]].
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| For instance, if <math>M \approx \mathbf{Z} \oplus \mathbf{Z}/2</math>, and <math>(1,0), (0,1)</math> is one basis, then
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| <math>(1,1), (0,1)</math> is another basis, and the change of basis matrix <math>\begin{bmatrix}1 & 1 \\0 & 1\end{bmatrix}</math> does not preserve the summand <math>\mathbf{Z}</math>. However, it does preserve the <math>\mathbf{Z}/2</math> summand, as this is the torsion submodule (equivalently here, the 2-torsion elements).
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| ==Generalizations==
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| ===Groups===
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| The [[Jordan–Hölder theorem]] is a more general result for finite groups (or modules over an arbitrary ring). In this generality, one obtains a [[composition series]], rather than a [[direct sum of modules|direct sum]].
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| The [[Krull–Schmidt theorem]] and related results give conditions under which a module has something like a primary decomposition, a decomposition as a direct sum of [[indecomposable module]]s in which the summands are unique up to order.
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| ===Primary decomposition===
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| The primary decomposition generalizes to finitely generated modules over commutative [[Noetherian ring]]s, and this result is called the [[Lasker–Noether theorem]].
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| ===Indecomposable modules===
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| By contrast, unique decomposition into ''indecomposable'' submodules does not generalize as far, and the failure is measured by the [[ideal class group]], which vanishes for PIDs.
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| For rings that are not principal ideal domains, unique decomposition need not even hold for modules over a ring generated by two elements. For the ring R = Z[√−5], both the module R and its submodule M generated by 2 and 1 + √−5 are indecomposable. While R is not isomorphic to M, R ⊕ R is isomorphic to M ⊕ M; thus the images of the M summands give indecomposable submodules L<sub>1</sub>, L<sub>2</sub> < R ⊕ R which give a different decomposition of R ⊕ R. The failure of uniquely factorizing R ⊕ R into a direct sum of indecomposable modules is directly related (via the ideal class group) to the failure of the unique factorization of elements of R into irreducible elements of R.
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| ===Non-finitely generated modules===
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| Similarly for modules that are not finitely generated, one cannot expect such a nice decomposition: even the number of factors may vary. There are '''Z'''-submodules of '''Q'''<sup>4</sup> which are simultaneously direct sums of two indecomposable modules and direct sums of three indecomposable modules, showing the analogue of the primary decomposition cannot hold for infinitely generated modules, even over the integers, '''Z'''.
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| Another issue that arises with non-finitely generated modules is that there are torsion-free modules which are not free. For instance, consider the ring '''Z''' of integers. Then '''Q''' is a torsion-free '''Z'''-module which is not free. Another classical example of such a module is the [[Baer–Specker group]], the group of all sequences of integers under termwise addition. In general, the question of which infinitely generated torsion-free abelian groups are free depends on which [[large cardinal]]s exist. A consequence is that any structure theorem for infinitely generated modules depends on a choice of set theory axioms and may be invalid under a different choice.
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| ==References==
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| {{reflist}}
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| {{refbegin}}
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| *{{Citation | last1=Dummit | first1=David S. | last2=Foote | first2=Richard M. | title=Abstract algebra | publisher=Wiley | location=New York | edition=3rd | isbn=978-0-471-43334-7 |mr=2286236 | year=2004}}
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| *{{Citation | last=Hungerford | first1=Thomas W. | author1-link=Thomas W. Hungerford | title=Algebra | publisher=Springer | location=New York | isbn=978-0-387-90518-1 | year=1980 | pages=218–226, Section IV.6: Modules over a Principal Ideal Domain }}
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| *{{Citation |author=Jacobson, Nathan |author1-link=Nathan Jacobson|title=Basic algebra. I |edition=2 |publisher=W. H. Freeman and Company |place=New York |date=1985 |pages=xviii+499 |isbn=0-7167-1480-9 |mr=780184}}
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| *{{Citation | last1=Lam | first1=T. Y. | title=Lectures on modules and rings | publisher=Springer-Verlag | series=Graduate Texts in Mathematics No. 189 | isbn=978-0-387-98428-5 | year=1999}}
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| {{refend}}
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| {{DEFAULTSORT:Structure Theorem For Finitely Generated Modules Over A Principal Ideal Domain}}
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| [[Category:Theorems in abstract algebra]]
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| [[Category:Module theory]]
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| [[de:Hauptidealring#Moduln über Hauptidealringen]]
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