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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.

Statement

When a vector space over a field F has a finite generating set, then one may extract from it a basis consisting of a finite number n of vectors, and the space is therefore isomorphic to Fⁿ. 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 Rⁿ with n finite (to see this it suffices to construct the morphism that sends the elements of the canonical basis Rⁿ to the generators of the module, and take the quotient by its kernel.) By changing the choice of generating set, one can in fact describe the module as the quotient of some Rⁿ by a particularly simple submodule, and this is the structure theorem.

The structure theorem for finitely generated modules over a principal ideal domain usually appears in the following two forms.

Invariant factor decomposition

Every finitely generated module M over a principal ideal domain R is isomorphic to a unique one of the form

⨁_{i} R / (d_{i}) = R / (d_{1}) \oplus R / (d_{2}) \oplus \dots \oplus R / (d_{n})

where $(d_{i}) \neq R$ and $d_{i} | d_{i + 1}$ . The order of the nonzero $(d_{i}) \neq R$ ideals is invariant, and the number of $(d_{i}) = 0$ is invariant.

The nonzero $d_{i}$ elements, together with the number of $d_{i}$ 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. The $R / (d_{i})$ themselves are called invariant factors of M.

The ideals $(d_{i})$ are unique. In terms of the $d_{i}$ elements, this means that the $d_{i}$ are unique up to multiplication by a unit.

The free part is visible in the part of the decomposition corresponding to the $d_{i} = 0$ factors. These occur at the end of the sequence of $d_{i}$ 's, as everything divides zero.

Some prefer to write the free part of M separately:

R^{f} \oplus ⨁_{i} R / (d_{i}) = R^{f} \oplus R / (d_{1}) \oplus R / (d_{2}) \oplus \dots \oplus R / (d_{n - f})

where the visible $d_{i}$ are nonzero, and f is the number of $d_{i}$ 's which are 0.

Primary decomposition

Every finitely generated module M over a principal ideal domain R is isomorphic to one of the form

⨁_{i} R / (q_{i})

where

(q_{i}) \neq R

and the

(q_{i})

are primary ideals. The

q_{i}

are unique (up to multiplication by units).

The elements $q_{i}$ are called the elementary divisors of M. In a PID, primary ideals are powers of primes, and so $(q_{i}) = (p_{i}^{r_{i}}) = (p_{i})^{r_{i}}$ .

The summands $R / (q_{i})$ are indecomposable, so the primary decomposition is a decomposition into indecomposable modules, and thus every finitely generated module over a PID is a completely decomposable module. Since PID's are Noetherian rings, this can be seen as a manifestation of the Lasker-Noether theorem.

As before, it is possible to write the free part (where $q_{i} = 0$ ) separately and express M as:

R^{f} \oplus (⨁_{i} R / (q_{i}))

where the visible $q_{i}$ are nonzero.

Proofs

One proof proceeds as follows:

Every finitely generated module over a PID is also finitely presented because a PID is Noetherian, an even stronger condition than coherence.
Take a presentation, which is a map $R^{r} \to R^{g}$ (relations to generators), and put it in Smith normal form.

This yields the invariant factor decomposition, and the diagonal entries of Smith normal form are the invariant factors.

Another outline of a proof:

Denote by tM the torsion submodule of M. Then M/tM is a finitely generated torsion free module, and such a module over a commutative PID is a free module of finite rank, so it is isomorphic to $R^{n}$ 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 $M = t M \oplus F$ .
For a prime p in R we can then speak of $N_{p} = {m \in t M ∣ \exists i, m p^{i} = 0}$ for each prime p. This is a submodule of tM, and it turns out that each N_p is a direct sum of cyclic modules, and that tM is a direct sum of N_p for a finite number of distinct primes p.
Putting the previous two steps together, M is decomposed into cyclic modules of the indicated types.

Corollaries

This includes the classification of finite-dimensional vector spaces as a special case, where $R = K$ . Since fields have no non-trivial ideals, every finitely generated vector space is free.

Taking $R = ℤ$ yields the fundamental theorem of finitely generated abelian groups.

Let T be a linear operator on a finite-dimensional vector space V over K. Taking $R = K [T]$ , 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 $K [T]$ . The last invariant factor is the minimal polynomial, and the product of invariant factors is the characteristic polynomial. Combined with a standard matrix form for $K [T] / p (T)$ , this yields various canonical forms:

invariant factors + companion matrix yields Frobenius normal form (aka, rational canonical form)
primary decomposition + companion matrix yields primary rational canonical form
primary decomposition + Jordan blocks yields Jordan canonical form (this latter only holds over an algebraically closed field)

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 decomposition. This follows because there are non-trivial automorphisms of these modules which do not preserve the summands.

However, one has a canonical torsion submodule T, and similar canonical submodules corresponding to each (distinct) invariant factor, which yield a canonical sequence:

0 < \dots < T < M .

Compare composition series in Jordan–Hölder theorem.

For instance, if $M \approx Z \oplus Z / 2$ , and $(1, 0), (0, 1)$ is one basis, then $(1, 1), (0, 1)$ is another basis, and the change of basis matrix $[\begin{matrix} 1 & 1 \\ 0 & 1 \end{matrix}]$ does not preserve the summand $Z$ . However, it does preserve the $Z / 2$ summand, as this is the torsion submodule (equivalently here, the 2-torsion elements).

Generalizations

Groups

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.

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 modules in which the summands are unique up to order.

Primary decomposition

The primary decomposition generalizes to finitely generated modules over commutative Noetherian rings, and this result is called the Lasker–Noether theorem.

Indecomposable modules

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.

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 L₁, L₂ < 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.

Non-finitely generated modules

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 Q⁴ 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.

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 cardinals 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.

References

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15.6.1 As the agent is an intermediary, generally, as soon as the principal and third party are introduced right into a contractual relationship, the agent drops out of the image, subject to any problems with remuneration or indemnification that he could have against the principal, and extra exceptionally, against the third occasion. Generally, agents are entitled to be indemnified for all liabilities reasonably incurred within the execution of the brokers´ authority.

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Many property agents need to declare for the PIC grant in Singapore. However, not all of them know find out how to do the correct process for getting this PIC scheme from the IRAS. There are a number of steps that you need to do before your software can be approved.

Naturally, you will have to pay a safety deposit and that is usually one month rent for annually of the settlement. That is the place your good religion deposit will likely be taken into account and will kind part or all of your security deposit. Anticipate to have a proportionate amount deducted out of your deposit if something is discovered to be damaged if you move out. It's best to you'll want to test the inventory drawn up by the owner, which can detail all objects in the property and their condition. If you happen to fail to notice any harm not already mentioned within the inventory before transferring in, you danger having to pay for it yourself.

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Have handed an trade examination i.e Widespread Examination for House Brokers (CEHA) or Actual Property Agency (REA) examination, or equal; Exclusive brokers are extra keen to share listing information thus making certain the widest doable coverage inside the real estate community via Multiple Listings and Networking. Accepting a severe provide is simpler since your agent is totally conscious of all advertising activity related with your property. This reduces your having to check with a number of agents for some other offers. Price control is easily achieved. Paint work in good restore-discuss with your Property Marketing consultant if main works are still to be done. Softening in residential property prices proceed, led by 2.8 per cent decline within the index for Remainder of Central Region

Once you place down the one per cent choice price to carry down a non-public property, it's important to accept its situation as it is whenever you move in – faulty air-con, choked rest room and all. Get round this by asking your agent to incorporate a ultimate inspection clause within the possibility-to-buy letter. HDB flat patrons routinely take pleasure in this security net. "There's a ultimate inspection of the property two days before the completion of all HDB transactions. If the air-con is defective, you can request the seller to repair it," says Kelvin.

15.6.1 As the agent is an intermediary, generally, as soon as the principal and third party are introduced right into a contractual relationship, the agent drops out of the image, subject to any problems with remuneration or indemnification that he could have against the principal, and extra exceptionally, against the third occasion. Generally, agents are entitled to be indemnified for all liabilities reasonably incurred within the execution of the brokers´ authority.

To achieve the very best outcomes, you must be always updated on market situations, including past transaction information and reliable projections. You could review and examine comparable homes that are currently available in the market, especially these which have been sold or not bought up to now six months. You'll be able to see a pattern of such report by clicking here It's essential to defend yourself in opposition to unscrupulous patrons. They are often very skilled in using highly unethical and manipulative techniques to try and lure you into a lure. That you must also protect your self, your loved ones, and personal belongings as you'll be serving many strangers in your home. Sign a listing itemizing of all of the objects provided by the proprietor, together with their situation. HSR Prime Recruiter 2010
Many property agents need to declare for the PIC grant in Singapore. However, not all of them know find out how to do the correct process for getting this PIC scheme from the IRAS. There are a number of steps that you need to do before your software can be approved.

Naturally, you will have to pay a safety deposit and that is usually one month rent for annually of the settlement. That is the place your good religion deposit will likely be taken into account and will kind part or all of your security deposit. Anticipate to have a proportionate amount deducted out of your deposit if something is discovered to be damaged if you move out. It's best to you'll want to test the inventory drawn up by the owner, which can detail all objects in the property and their condition. If you happen to fail to notice any harm not already mentioned within the inventory before transferring in, you danger having to pay for it yourself.

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de:Hauptidealring#Moduln über Hauptidealringen

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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.
+==Statement==
+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.
+The structure theorem for [[finitely generated module]]s over a [[principal ideal domain]] usually appears in the following two forms.
+===Invariant factor decomposition===
+Every finitely generated module ''M'' over a principal ideal domain ''R'' is isomorphic to a unique one of the form
+:<math>\bigoplus_i R/(d_i) = R/(d_1)\oplus R/(d_2)\oplus\cdots\oplus R/(d_n)</math>
+where <math>(d_i) \neq R</math> and <math>d_i \vert d_{i+1}</math>. The order of the nonzero <math>(d_i) \neq R</math> ideals is invariant, and the number of <math>(d_i)=0</math> is invariant.
+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. The <math>R/(d_i)</math> themselves are called [[invariant factor]]s of ''M''.
+The ideals <math>(d_i)</math> are unique. In terms of the <math>d_i</math> elements, this means that the <math>d_i</math> are unique up to multiplication by a [[unit (ring theory)|unit]].
+The free part is visible in the part of the decomposition corresponding to the <math>d_i = 0</math> factors. These occur at the end of the sequence of <math>d_i</math>'s, as everything divides zero.
+Some prefer to write the free part of ''M'' separately:
+:<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>
+where the visible <math>d_i</math> are nonzero, and ''f'' is the number of <math>d_i</math>'s which are 0.
+===Primary decomposition===
+:Every [[finitely generated module]] ''M'' over a [[principal ideal domain]] ''R'' is isomorphic to one of the form
+::<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).
+The elements <math>q_i</math> are called the ''elementary divisors'' of ''M''. In a PID, primary ideals are powers of primes, and so <math>(q_i)=(p_i^{r_i}) = (p_i)^{r_i}</math>.
+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]].
+As before, it is possible to write the free part (where <math>q_i=0</math>) separately and express ''M'' as:
+:<math>R^f \oplus(\bigoplus_i R/(q_i))</math>
+where the visible <math>q_i </math> are nonzero.
+==Proofs==
+One proof proceeds as follows:
+* 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]].
+* Take a presentation, which is a map <math>R^r \to R^g</math> (relations to generators), and put it in [[Smith normal form]].
+This yields the invariant factor decomposition, and the diagonal entries of Smith normal form are the invariant factors.
+Another outline of a proof:
+* 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>.
+* 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''.
+* Putting the previous two steps together, ''M'' is decomposed into cyclic modules of the indicated types.
+<!-- (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]].
+* 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.
+* ''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.
+*: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).
+* 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.
+* 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]].-->
+==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.
+Taking <math>R=\mathbb{Z}</math> yields the [[fundamental theorem of finitely generated abelian groups]].
+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:
+* [[invariant factors]] + [[companion matrix]] yields [[Frobenius normal form]] (aka, [[rational canonical form]])
+* [[primary decomposition]] + [[companion matrix]] yields [[primary rational canonical form]]
+* [[primary decomposition]] + [[Jordan block]]s yields [[Jordan canonical form]] (this latter only holds over an [[algebraically closed field]])
+==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.
+However, one has a canonical torsion submodule ''T'', and similar canonical submodules corresponding to each (distinct) invariant factor, which yield a canonical sequence:
+:<math>0 < \cdots < T < M.</math>
+Compare [[composition series]] in [[Jordan–Hölder theorem]].
+For instance, if <math>M \approx \mathbf{Z} \oplus \mathbf{Z}/2</math>, and <math>(1,0), (0,1)</math> is one basis, then
+<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).
+==Generalizations==
+===Groups===
+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]].
+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.
+===Primary decomposition===
+The primary decomposition generalizes to finitely generated modules over commutative [[Noetherian ring]]s, and this result is called the [[Lasker–Noether theorem]].
+===Indecomposable modules===
+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.
+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&nbsp;=&nbsp;Z[√−5], both the module R and its submodule M generated by 2 and 1&nbsp;+&nbsp;√−5 are indecomposable. While R is not isomorphic to M, R&nbsp;⊕&nbsp;R is isomorphic to M&nbsp;⊕&nbsp;M; thus the images of the M summands give indecomposable submodules L<sub>1</sub>,&nbsp;L<sub>2</sub>&nbsp;<&nbsp;R&nbsp;⊕&nbsp;R which give a different decomposition of R&nbsp;⊕&nbsp;R. The failure of uniquely factorizing R&nbsp;⊕&nbsp;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&nbsp;R.
+===Non-finitely generated modules===
+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'''.
+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.
+==References==
+{{reflist}}
+{{refbegin}}
+*{{Citation | last1=Atiyah | first1=Michael Francis | author1-link=Michael Atiyah | last2=Macdonald | first2=I.G. | author2-link=Ian G. Macdonald | title=Introduction to Commutative Algebra | publisher=Westview Press | isbn=978-0-201-40751-8 | year=1969}}
+*{{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 | id={{MathSciNet | id = 2286236}} | year=2004}}
+*{{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 }}
+*{{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}}
+*{{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}}
+{{refend}}
+{{DEFAULTSORT:Structure Theorem For Finitely Generated Modules Over A Principal Ideal Domain}}
+[[Category:Theorems in abstract algebra]]
+[[Category:Module theory]]
+[[de:Hauptidealring#Moduln über Hauptidealringen]]

Trimean: Difference between revisions

Latest revision as of 17:18, 17 October 2013

Contents

Statement

Invariant factor decomposition

Primary decomposition

Proofs

Corollaries

Uniqueness

Generalizations

Groups

Primary decomposition

Indecomposable modules

Non-finitely generated modules

References

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Trimean: Difference between revisions

Latest revision as of 17:18, 17 October 2013

Statement

Invariant factor decomposition

Primary decomposition

Proofs

Corollaries

Uniqueness

Generalizations

Groups

Primary decomposition

Indecomposable modules

Non-finitely generated modules

References

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