# Main ideal ring

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:

• 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:

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 on = 0}$${\ displaystyle P \ subset A}$${\ displaystyle M}$${\ displaystyle p}$${\ displaystyle M _ {(p)}}$

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

With

${\ displaystyle M _ {(p)} = \ left \ {m \ in M ​​\ mid p ^ {i} m = 0 \ {\ text {for a}} \ i \ in \ mathbb {N} \ right \} .}$

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).