Main ideal ring

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In algebra , a branch of mathematics , integrity rings are referred to as main ideal rings or main ideal domains if each ideal is a main ideal . The most important examples of main ideal rings are the ring of whole numbers and polynomial rings in an indeterminate over a field . The concept of the main ideal ring allows statements about these two special cases to be formulated uniformly. Examples for applications of the general theory are the Jordan normal form , the partial fraction decomposition or the structure theory of finitely generated Abelian groups .

definition

An integrity ring (i.e. a commutative ring with zero divisors ) is called a principal ideal ring if every ideal is a principal ideal ; H. there is one such that . ${\displaystyle A}$ ${\displaystyle 1\neq 0}$ ${\displaystyle I\subseteq A}$ ${\displaystyle x\in A}$ ${\displaystyle I=A\cdot x=\left\{a\cdot x\mid a\in A\right\}}$

In the following, let a main ideal ring and its quotient field be . In addition, let it be a set that contains exactly one associated element for each irreducible . In the case , the set of (positive) prime numbers is such , in the case for a field, the set of irreducible polynomials with leading coefficient 1. ${\displaystyle A}$ ${\displaystyle K}$ ${\displaystyle P\subset A}$ ${\displaystyle p\in A}$ ${\displaystyle p}$ ${\displaystyle A=\mathbb {Z} }$ ${\displaystyle P}$ ${\displaystyle A=k[T]}$ ${\displaystyle k}$

Examples, conclusions and counterexamples

The following rings are main ideal rings:

• body
• ${\displaystyle \mathbb {Z} }$(the ring of whole numbers )
• ${\displaystyle \mathbb {Z} [i]}$(the ring of whole Gaussian numbers )
• Polynomial rings in an indeterminate over a field ${\displaystyle k[T]}$ ${\displaystyle k}$
• formal power series rings in an indeterminate over a body ${\displaystyle k[[T]]}$ ${\displaystyle k}$
• discrete rating rings
• Euclidean rings (although this class includes all of the preceding examples, not every main ideal ring is Euclidean)
• Localizations of main ideal rings are again main ideal rings.
• The wholeness ring of the body , i.e. H. the ring of the Eisenstein numbers is a main ideal ring. The following statement even applies: The totality ring of a square number field with a negative, square-free one is a main ideal ring if and only if the proof is based on the investigation of the ideal class group , which in number fields can be seen as a measure of how far a ring is from it, to be a main ideal ring. ${\displaystyle \mathbb {Q} ({\sqrt {-3}})}$ ${\displaystyle K=\mathbb {Q} ({\sqrt {d}})}$ ${\displaystyle d\in \mathbb {Z} }$ ${\displaystyle d\in \lbrace -1,-2,-3,-7,-11,-19,-43,-67,-163\rbrace .}$

Main ideal rings belong to the following more general classes of rings:

• factorial rings . In particular:

• An element is prime if and only if it is irreducible . ${\displaystyle a\in A\setminus \{0\}}$
• Each non-zero element of the quotient field of can be expressed in a unique way in the form ${\displaystyle A}$

${\displaystyle u\cdot \prod _{p\in P}p^{e_{p}}}$

write with whole numbers and a unit . ${\displaystyle e_{p}}$ ${\displaystyle u\in A^{\times }}$

• The lemma of Gauss : Every irreducible element in is either an irreducible element of (understood as a constant polynomial) or an irreducible polynomial in whose coefficients are relatively prime. ${\displaystyle A[X]}$ ${\displaystyle A}$ ${\displaystyle K[X]}$

• Main ideal rings are trivially Noetherian rings, since every ideal is finitely generated (by an element).
• The main ideal rings are always Dedekind rings (see also below )

No main ideal rings are:

• The polynomial ring over the integers is not a main ideal ring because the ideal generated by and cannot be generated by a single polynomial. According to Gauss's lemma, this ring is factorial because it is a polynomial ring over a factorial ring. ${\displaystyle \mathbb {Z} [X]}$ ${\displaystyle 2}$ ${\displaystyle X}$
• The ring is not a main ideal ring because the ideal is not a main ideal. ${\displaystyle k[x,y]}$ ${\displaystyle (x,y)}$
• The ring is not a main ideal ring as it is not an integrity ring. But every ideal in this ring is a main ideal. ${\displaystyle \mathbb {Z} /4\mathbb {Z} }$

Divisibility

• The (up to Associated unit unique) greatest common divisor of elements is the (up to unique Associated slope) of the ideal generator . In particular, Bézout's lemma applies : There exist with ${\displaystyle x_{1},\dots ,x_{m}}$ ${\displaystyle (x_{1},\dots ,x_{m})}$ ${\displaystyle a_{1},\dots ,a_{m}\in A}$

${\displaystyle \operatorname {ggT} (x_{1},\dots ,x_{m})=a_{1}x_{1}+\dots +a_{m}x_{m}.}$

Special case: are coprime if and only if there is with ${\displaystyle x_{1},\dots ,x_{k}}$ ${\displaystyle a_{1},\dots ,a_{m}}$

${\displaystyle 1=a_{1}x_{1}+\dots +a_{m}x_{m}.}$

• The least common multiple of is the generator of the ideal . ${\displaystyle x_{1},\ldots ,x_{m}}$ ${\displaystyle (x_{1})\cap \ldots \cap (x_{m})}$
• Chinese remainder of the sentence : If pairs are coprime, then the canonical ring homomorphism is ${\displaystyle x_{1},\dots ,x_{m}}$

${\displaystyle A/(x_{1}\cdots x_{m})\to \prod _{i=1}^{m}A/(x_{i})}$

an isomorphism.

• A tightening of the Chinese remainder theorem is the approximation theorem : Given , different pairs and numbers . Then there is a , the relative in approximated th order and is otherwise regularly, d. H. ${\displaystyle x_{1},\dots ,x_{m}\in K}$ ${\displaystyle p_{1},\dots ,p_{m}\in P}$ ${\displaystyle n_{1},\dots ,n_{m}\in \mathbb {N} }$ ${\displaystyle x\in K}$ ${\displaystyle x_{i}}$ ${\displaystyle p_{i}}$ ${\displaystyle n_{i}}$

${\displaystyle v_{p_{i}}(x-x_{i})\geq n_{i}}$ For ${\displaystyle i=1,\dots ,m}$

and

${\displaystyle v_{p}(x)\geq 0}$for . ${\displaystyle p\in P\setminus \{p_{1},\dots ,p_{m}\}}$

The exponent of denotes in the prime factorization of . ${\displaystyle v_{p}(x)\in \mathbb {Z} }$ ${\displaystyle p}$ ${\displaystyle x}$

• For are equivalent: ${\displaystyle p\in A\setminus \{0\}}$
  • ${\displaystyle p}$ is irreducible
  • ${\displaystyle p}$ is a prime element
  • ${\displaystyle (p)}$is a prime ideal
  • ${\displaystyle (p)}$is a maximum ideal

The zero ideal is also a prime ideal, but only maximum when there is a body. ${\displaystyle A}$

Main ideal rings as Dedekind rings

Main article: Dedekind ring

Many rings that occur naturally in algebraic number theory and algebraic geometry are not main ideal rings, but belong to a more general class of rings, the Dedekind rings. They are the localized version of the main ideal rings, ideals are no longer global, but only generated locally by an element:

If there is a Noetherian integrity domain for which the local ring is a main ideal ring for every prime ideal, then it is called a Dedekind ring . ${\displaystyle A}$ ${\displaystyle A_{\mathfrak {p}}}$ ${\displaystyle {\mathfrak {p}}}$ ${\displaystyle A}$

The following properties apply to main ideal rings, but also more generally to Dedekind rings:

• They are either solid or one-dimensional , i.e. H. every prime ideal unequal is maximal . ${\displaystyle (0)}$
• They are completely closed in their quotient body.
• You are regular .
• Your local rings are either solid or discrete rating rings .
• the above approximation theorem

If a Dedekind ring is factorial or semilocal , it is a main ideal ring.

Modules over main ideal rings

General

• Sub- modules of free modules are free.
• If there is a finitely generated module with a torsion sub- module , there is a free sub-module so that . Torsion-free, finitely generated modules are free. ${\displaystyle M}$ ${\displaystyle T}$ ${\displaystyle F\subseteq M}$ ${\displaystyle M=F\oplus T}$
• Projective modules are free.
• A module is injective if and only if it can be divided . Injective module quotients are injective, each module has an injective resolution of length 1. An explicit injective resolution of is ${\displaystyle A}$

${\displaystyle 0\to A\to K\to K/A\to 0.}$

Finally generated modules: elementary part replacement

The elementary substitute describes the structure of a decomposition of a finitely generated module into indivisible modules. (A module is called indivisible if there are no modules with .) ${\displaystyle M}$ ${\displaystyle M_{1},M_{2}\neq 0}$ ${\displaystyle M\cong M_{1}\oplus M_{2}}$

As above, let it be a system of representatives of the irreducible elements (except for association). For every finitely generated module there are uniquely certain non-negative integers and for , almost all of which are zero, so that ${\displaystyle P}$ ${\displaystyle M}$ ${\displaystyle m_{0}}$ ${\displaystyle m_{p,i}}$ ${\displaystyle p\in P,i\in \mathbb {N} _{\geq 1}}$

${\displaystyle M\cong A^{m_{0}}\oplus \bigoplus _{p\in P}\bigoplus _{i\geq 1}(A/(p^{i}))^{m_{p,i}}.}$

The numbers are clearly defined by, and the individual factors or are indivisible. The ideals for which applies are called elementary divisors of . ${\displaystyle m_{0},m_{p,i}}$ ${\displaystyle M}$ ${\displaystyle A}$ ${\displaystyle A/(p^{k})}$ ${\displaystyle (p^{i})}$ ${\displaystyle m_{p,i}\neq 0}$ ${\displaystyle M}$

Finally generated modules: invariant factors

For every finitely generated module there is a finite sequence of elements of which are not necessarily different from zero such that ${\displaystyle M}$ ${\displaystyle x_{1},x_{2},\dots ,x_{m}}$ ${\displaystyle A}$

• ${\displaystyle x_{i}\mid x_{i+1}}$ For ${\displaystyle i=1,2,\dots ,m-1}$
• ${\displaystyle M\cong \bigoplus _{i=1}^{m}A/(x_{i}).}$

The ideals are uniquely determined by and are called the invariant factors of . The elements are consequently clearly defined except for association. ${\displaystyle (x_{i})}$ ${\displaystyle M}$ ${\displaystyle M}$ ${\displaystyle x_{i}}$

There are two competing perspectives on this statement about modules:

• You can choose producers for a module and consider the core of the associated homomorphism . ${\displaystyle M}$ ${\displaystyle w_{1},\dots ,w_{m}}$ ${\displaystyle U\subseteq A^{m}}$ ${\displaystyle A^{m}\to M}$
• For a sub-module one can choose producer and look at the matrix with entries in , which describes the homomorphism with image . ${\displaystyle U\subseteq A^{m}}$ ${\displaystyle u_{1},\dots ,u_{n}}$ ${\displaystyle m\times n}$ ${\displaystyle X}$ ${\displaystyle A}$ ${\displaystyle A^{n}\to A^{m}}$ ${\displaystyle U}$

The reverse is the image of a matrix with entries in a sub-module , and the quotient module (the coke of the homomorphism given by ) is a finitely generated module. ${\displaystyle m\times n}$ ${\displaystyle A}$ ${\displaystyle U\subseteq A^{m}}$ ${\displaystyle M=A^{m}/U}$ ${\displaystyle X}$ ${\displaystyle A^{n}\to A^{m}}$ ${\displaystyle A}$

For sub-modules of free modules the statement reads:

• If a free module and a (also free) sub-module of is of rank , there are elements that are part of a base of , as well as elements with such that a base of is. The part spanned by the can be invariantly described as the archetype of the torsion sub-module of . The ideals are the invariants (as above) of the module , possibly supplemented by . ${\displaystyle F}$ ${\displaystyle A}$ ${\displaystyle U}$ ${\displaystyle F}$ ${\displaystyle r}$ ${\displaystyle n}$ ${\displaystyle e_{1},\dots ,e_{r}\in F}$ ${\displaystyle F}$ ${\displaystyle x_{1},\dots ,x_{r}\in A}$ ${\displaystyle x_{1}\mid x_{2}\mid \dots \mid x_{r}}$ ${\displaystyle x_{1}e_{1},\dots ,x_{r}e_{r}}$ ${\displaystyle U}$ ${\displaystyle e_{k}}$ ${\displaystyle F'\subseteq F}$ ${\displaystyle F/U}$ ${\displaystyle (x_{k})}$ ${\displaystyle F'/U}$ ${\displaystyle x_{k+1}=\dots =x_{m}=0}$

For matrices ( Smith normal form ):

• If a matrix is ​​of rank with entries in , there are invertible matrices such that it has the following form: ${\displaystyle X}$ ${\displaystyle m\times n}$ ${\displaystyle r}$ ${\displaystyle A}$ ${\displaystyle P\in \operatorname {GL} (m,A),Q\in \operatorname {GL} (n,A)}$ ${\displaystyle PXQ}$

${\displaystyle {\begin{pmatrix}x_{1}&0&\cdots &0&0&\cdots &0\\0&x_{2}&\ddots &\vdots &\vdots &&\vdots \\\vdots &\ddots &\ddots &0&\vdots &&\vdots \\0&\cdots &0&x_{r}&0&\cdots &0\\0&\cdots &\cdots &0&0&\cdots &0\\\vdots &&&\vdots &\vdots &&\vdots \\0&\cdots &\cdots &0&0&\cdots &0\end{pmatrix}}}$

The invariants are again as above. ${\displaystyle x_{1}\mid x_{2}\mid \dots \mid x_{r}}$

Torsional modules

Let there be a (not necessarily finitely generated) torsional modulus over , i.e. H. for each there is a with . Again let a representative system of the irreducible elements. Then: is the direct sum of the -primary sub-modules , i.e. H. ${\displaystyle M}$ ${\displaystyle A}$ ${\displaystyle m\in M}$ ${\displaystyle a\in A\setminus \{0\}}$ ${\displaystyle am=0}$ ${\displaystyle P\subset A}$ ${\displaystyle M}$ ${\displaystyle p}$ ${\displaystyle M_{(p)}}$

${\displaystyle M=\bigoplus _{p\in P}M_{(p)}}$

With

${\displaystyle M_{(p)}=\left\{m\in M\mid p^{i}m=0\ {\text{für ein}}\ i\in \mathbb {N} \right\}.}$

The corollary shows that it is semi-easy if and only if for all . ${\displaystyle M}$ ${\displaystyle p\cdot M_{(p)}=0}$ ${\displaystyle p\in P}$

Application examples:

• If and , the statement reads: Every rational number has a unique representation ${\displaystyle A=\mathbb {Z} }$ ${\displaystyle M=K/A=\mathbb {Q} /\mathbb {Z} }$

${\displaystyle a+\sum _{p\ {\text{prim}}}\sum _{i=1}^{o_{p}}d_{p,i}p^{-i}}$

with , (and almost all ) as well as and . ${\displaystyle a\in \mathbb {Z} }$ ${\displaystyle o_{p}\geq 0}$ ${\displaystyle o_{p}=0}$ ${\displaystyle d_{p,i}\in \{0,1,\dots ,p-1\}}$ ${\displaystyle d_{p,o_{p}}\neq 0}$

• Is ( a body) and , then corresponds to the rational functions whose denominator is a power of . The theorem provides the first step in the partial fraction decomposition , i.e. H. the unambiguous representation of a rational function as ${\displaystyle A=k[T]}$ ${\displaystyle k}$ ${\displaystyle M=K/A=k(T)/k[T]}$ ${\displaystyle M_{(p)}}$ ${\displaystyle p}$

${\displaystyle a+\sum _{p}\sum _{i=1}^{o_{p}}d_{p,i}p^{-i}.}$

The irreducible normalized polynomials in runs through , the other components are the regular part , the orders (almost all ) and suitable polynomials for with . In particular , if linear, then are the constants. ${\displaystyle p}$ ${\displaystyle k[T]}$ ${\displaystyle a\in k[T]}$ ${\displaystyle o_{p}\geq 0}$ ${\displaystyle o_{p}=0}$ ${\displaystyle d_{p,i}}$ ${\displaystyle i=1,2,\dots ,o_{p}}$ ${\displaystyle \deg(d_{p,i})<\deg(p)}$ ${\displaystyle p}$ ${\displaystyle d_{p,i}}$

• If and is a finite-dimensional vector space together with an endomorphism (with the module structure ), then the above decomposition is the split into the main spaces . In this case, the corollary says that is semi-simple if and only if the minimal polynomial of does not contain any multiple factors. ${\displaystyle A=k[T]}$ ${\displaystyle M}$ ${\displaystyle k}$ ${\displaystyle f}$ ${\displaystyle A}$ ${\displaystyle Tv=f(v)}$ ${\displaystyle f}$ ${\displaystyle f}$

Generalization to non-commutative rings

The definitions can be generalized to non-commutative rings. A right main ideal is a right multiple of a single element ; is a left main ideal. As in the commutative case, and are the trivial (and bilateral) main ideals. ${\displaystyle I}$ ${\displaystyle gA}$ ${\displaystyle g\in A}$ ${\displaystyle Ag}$ ${\displaystyle \{0\}=0A=A0}$ ${\displaystyle A=1A=A1}$

The Hurwitzquaternionen are an example of a non-commutative ring with its standard as is the left side as the Euclidean norm both left and rechtseuklidisch and thus both right a principal ideal.

Related terms

• If only it is demanded that every ideal is finite, one arrives at the concept of Noetherian ring .
• Conversely, one can set the condition for an integrity domain that all finitely generated ideals are main ideals: These are the so-called Bézout rings . The main ideal rings are precisely the Noetherian Bézout rings.
• Sometimes rings that are not free of zero divisors are also allowed in the definition of the term “main ideal ring”, so it is only required that every ideal is a main ideal and . In English, a distinction is made between the principal ideal ring and the principal ideal domain ( domain = integrity area). The corresponding distinction between the terms main ideal ring and main ideal area is unusual in German. ${\displaystyle 1\neq 0}$

literature

• Serge Lang : Algebra. Revised 3rd edition. Springer, Berlin et al. 2002, ISBN 0-387-95385-X ( Graduate Texts in Mathematics 211).
• Nicolas Bourbaki : Algebra II. Chapters 4-7. Springer, Berlin et al. 1990, ISBN 3-540-19375-8 ( Elements of Mathematics ).
• Nicolas Bourbaki: Eléments de mathématique. Algèbre Commutative. Volume 10: Chapitre 10. Réimpression de l'édition de 1998. Springer, Berlin et al. 2007, ISBN 978-3-540-34394-3 .
• Nicolas Bourbaki: Commutative Algebra. Chapters 1-7. 2nd printing. Springer, Berlin et al. 1989, ISBN 3-540-19371-5 ( Elements of Mathematics ).
• Stefan Müller-Stach , Jens Piontkowski: Elementary and algebraic number theory. A modern approach to classic topics. Vieweg, Wiesbaden 2006, ISBN 3-8348-0211-5 ( Vieweg studies ).

Individual evidence

1. ↑ Lang, Theorem II.5.2, p 112
2. ↑ Long, Theorem IV.2.3, p 182
3. ↑ Long, Corollary II.2.2, p.95
4. ↑ Bourbaki, Commutative Algebra, Ch. VII, §2.4, Proposition 2
5. ↑ Bourbaki, Commutative Algebra, Ch. VII, §2
6. ↑ Stefan Müller-Stach, Jens Piontkowski: Elementary and algebraic number theory . Vieweg-Verlag, 2006, p. 188 (sentence 18.16)
7. ↑ Bourbaki, Algebra, Ch. VII, § 3, Corollary 2; Lang, Theorem III.7.1
8. ↑ Bourbaki, Algebra, Ch. VII, § 4, No. 4, Corollary 1 and 2; Lang, Theorem III.7.3
9. ↑ Bourbaki, Algebra, Ch. VII, § 3, Corollary 3
10. ↑ Bourbaki, Algèbre, Ch. X, § 1, No. 7, Corollaire 2
11. ↑ Bourbaki, Algebra, Ch. VII, § 4, No. 8, Proposition 9; Lang, Theorem III.7.5
12. ↑ Bourbaki, Algebra, Ch. VII, § 4, No. 4, theorem 2; Lang, Theorem III.7.7
13. ↑ Bourbaki, Algebra, Ch. VII, § 4, No. 3, theorem 1; Lang, Theorem III.7.8
14. ↑ Bourbaki, Algebra, Ch. VII, § 4, No. 6, Corollary 1; Lang, Theorem III.7.9
15. ↑ Bourbaki, Algebra, Ch. VII, § 2, No. 2, theorem 1
16. ↑ Bourbaki, Algebra, Ch. VII, § 2, No. 2, Corollary 4
17. ↑ Bourbaki, Algebra, Ch. VII, § 2, No. 3, I.
18. ↑ Bourbaki, Algebra, Ch. VII, § 2, No. 3, II
19. ↑ Bourbaki, Algebra, Ch. VII, § 5, No. 8, proposition 14
20. ↑ Lang, II, §1, p. 86
21. ↑ Rainer Schulze-Pillot: Introduction to Algebra and Number Theory. Springer-Verlag, 2014, ISBN 978-3-642-55216-8 , p. 34 ( limited preview in the Google book search).