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 .


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 .

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.

Examples, conclusions and counterexamples

The following rings are main ideal rings:

  • body
  • (the ring of whole numbers )
  • (the ring of whole Gaussian numbers )
  • Polynomial rings in an indeterminate over a field
  • formal power series rings in an indeterminate over a body
  • 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.

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

  • An element is prime if and only if it is irreducible .
  • Each non-zero element of the quotient field of can be expressed in a unique way in the form
write with whole numbers and a unit .
  • 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.
  • 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.
  • The ring is not a main ideal ring because the ideal is not a main ideal.
  • 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.


  • 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
Special case: are coprime if and only if there is with
  • The least common multiple of is the generator of the ideal .
  • Chinese remainder of the sentence : If pairs are coprime, then the canonical ring homomorphism is
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.
for .
The exponent of denotes in the prime factorization of .
  • For are equivalent:
    • is irreducible
    • is a prime element
    • is a prime ideal
    • is a maximum ideal
The zero ideal is also a prime ideal, but only maximum when there is a body.

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 .

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


  • 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.
  • 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

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 .)

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

The numbers are clearly defined by, and the individual factors or are indivisible. The ideals for which applies are called elementary divisors of .

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

  • For

The ideals are uniquely determined by and are called the invariant factors of . The elements are consequently clearly defined except for association.

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 .
  • For a sub-module one can choose producer and look at the matrix with entries in , which describes the homomorphism with image .

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.

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 .

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:
The invariants are again as above.

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.


The corollary shows that it is semi-easy if and only if for all .

Application examples:

  • If and , the statement reads: Every rational number has a unique representation
with , (and almost all ) as well as and .
  • 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
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.
  • 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.

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.

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.


  • 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).