Irreducible ideal

An irreducible ideal in a commutative ring with is a real ideal that does not allow a non-trivial decomposition as the intersection of two other ideals. From the point of view of number theory , irreducible ideals represent a generalization of prime powers. Decomposing irreducible ideals therefore yield generalizations of the prime factorization and the decomposition into prime ideals in Dedekind rings . In Noether's rings a decomposition into irreducible ideals is at the same time a primary decomposition. In algebraic geometry , decomposing into irreducible ideals yields decompositions into topologically irreducible components . ${\ displaystyle 1}$

definition

A real ideal in a commutative ring with is called irreducible if it cannot be represented as the intersection of two really larger ideals. If there is such a representation, the ideal is called reducible. ${\ displaystyle 1}$

Examples

Prime ideals are irreducible ideals because their complements are multiplicatively closed.

In a Dedekind ring , an ideal is irreducible precisely because it is clearly broken down into prime ideals if it is the power of a prime ideal. Thus, in Dedekindringen, the irreducible ideals agree with the primary ideals . In particular, the terms irreducible and primary ideal coincide in main ideal rings and one obtains as irreducible ideals the ideals generated by the powers of irreducible elements and the null ideal.

In the polynomial ring over a body , a monomial ideal - i.e. an ideal that is generated by monomials - is irreducible if and only if it is generated by powers of variables. ${\ displaystyle K [X_ {1}, \ dotsc, X_ {n}]}$ ${\ displaystyle K}$

In the polynomial ring over a body , the ideal as the power of the maximum ideal is a primary ideal. However, it applies and is therefore not irreducible. ${\ displaystyle K [X, Y]}$ ${\ displaystyle K}$ ${\ displaystyle I = (X ^ {2}, XY, Y ^ {2})}$ ${\ displaystyle (X, Y)}$ ${\ displaystyle I = (X, Y ^ {2}) \ cap (X ^ {2}, Y)}$ ${\ displaystyle I}$

properties

A true ideal in a factor ring is irreducible if and only if its archetype is irreducible under canonical epimorphism. In particular, an ideal is irreducible if and only if the zero ideal in the associated factor ring is an irreducible ideal.

Connection with primary ideals

In Noether's rings every irreducible ideal is primary. In particular, the radical of an irreducible ideal is then a prime ideal, so that in the case of Noetherian rings we can speak of the prime ideal belonging to an irreducible ideal.

In non-Noetherian rings, irreducible ideals need not be primary. There are even non-Noetherian rings with irreducible ideals whose radical is not irreducible. However, for non-Noetherian rings there is at least an explicit characterization of the irreducible ideals, which are also primary.

Even in factorial Noetherian rings, not every primary ideal is irreducible, as the above example of the ideal shows. A -primary ideal of a Noetherian ring is irreducible if and only if there is no further primary ideal between and the ideal quotient . ${\ displaystyle (X ^ {2}, XY, Y ^ {2}) \ subseteq K [X, Y]}$ ${\ displaystyle P}$ ${\ displaystyle Q}$ ${\ displaystyle Q}$ ${\ displaystyle Q: P}$

Decomposing into irreducible ideals in Noetherian rings

In Noetherian rings, due to Noetherian induction, every real ideal can be represented as the intersection of a finite number of irreducible ideals. Since irreducible ideals are primary in Noetherian rings, this decomposition is a primary decomposition and thus in Noetherian rings in particular a primary decomposition exists.
With a minimal decomposition into irreducible ideals in a Noetherian ring, the number of components is clearly determined. In addition, the associated prime ideals of the components are just as clearly defined as the multiplicity in which they occur. The isolated irreducible components, whose prime ideals are minimally above the ideal, are also clearly determined.

Relation to other types of irreducibility

An irreducible element in a GCD ring is prime and therefore creates a prime ideal, which is in particular an irreducible ideal. However, the producer of an irreducible main ideal does not have to be an irreducible element, as the above examples in main ideal rings already show. Irreducible ideals therefore do not generalize irreducible elements, but their powers.

The zero set to an irreducible ideal of a polynomial ring over a field is an affine algebraic variety, that is a closed set with respect to the Zariski topology on which the subspace topology with respect to an irreducible topological space is. However, the sets of zeros of all ideals whose radical is prime are irreducible topological spaces. The decomposition of an ideal into irreducible ideals results in a decomposition of the set of zeros into irreducible components. ${\ displaystyle K [X_ {1}, \ dotsc, X_ {n}]}$ ${\ displaystyle K}$ ${\ displaystyle K ^ {n}}$

If any Noetherian commutative ring is included , the decomposition of an ideal into irreducible components yields a decomposition of the set of zeros in the spectrum of into finitely many topologically irreducible components with respect to the Zariski topology. ${\ displaystyle R}$ ${\ displaystyle 1}$ ${\ displaystyle I \ subseteq R}$ ${\ displaystyle I}$ ${\ displaystyle R}$

generalization

A real sub- module of a module is called irreducible if it cannot be written as the intersection of two really larger sub-modules. If we consider a ring as a module about itself, then the irreducible sub-modules are precisely the irreducible ideals. Irreducible sub-modules of Noether's modules are primary sub-modules and for real sub-modules of a Noether's module there is a decomposition into a finite number of irreducible modules, which generalizes the decomposition into irreducible ideals. Even with a very short breakdown into irreducible sub-modules, the number of irreducible components is clearly determined.

Further ideal classes, which, in addition to the irreducible and the primary ideals, also generalize the concept of prime power and provide finite decompositions of ideals, are quasi-primary ideals and primal ideals . Every irreducible ideal is both quasi-primary and primal.

Individual evidence

^ Michael Francis Atiyah, Ian Grant Macdonald: Introduction to Commutative Algebra. Addison-Wesley, Reading Mass. 1969, Proposition 1.11., P. 8
^ Michael Francis Atiyah, Ian Grant Macdonald: Introduction to Commutative Algebra. Addison-Wesley, Reading Mass. 1969, Theorem 9.3., P. 95.
^ Esra Miller, Bernd Sturmfels: Combinatorial Commutative Algebra. Springer, 2006, 5.16 f., 11.41.
^ Emmy Noether: Ideal theory in ring areas. In: Mathematical Annals. 83, 1921, §4, sentence VI., P. 39.
^ Michael Francis Atiyah, Ian Grant Macdonald: Introduction to Commutative Algebra. Addison-Wesley, Reading Mass. 1969, Lemma 7.12., P. 83.
↑ Is the radical of an irreducible ideal irreducible? mathoverflow. Retrieved December 8, 2019.
↑ Ladislas Fuchs: A condition under which an irreducible ideal is primary. In: The Quarterly Journal of Mathematics. Oxford Ser. 19 (1948), p. 235, Theorem 1
^ Michael Francis Atiyah, Ian Grant Macdonald: Introduction to Commutative Algebra. Addison-Wesley, Reading Mass. 1969, Exercises 3., p. 84.
↑ Wolfgang Gröbner: About irreducible ideals in commutative rings. In: Mathematical Annals. 110, 1935, sentence 4a. P. 205, GDZ
^ Emmy Noether: Ideal theory in ring areas. In: Mathematical Annals. 83, 1921, §2, sentence II., P. 33.
^ Michael Francis Atiyah, Ian Grant Macdonald: Introduction to Commutative Algebra. Addison-Wesley, Reading Mass. 1969, Lemma 7.11., P. 83.
^ Emmy Noether: Ideal theory in ring areas. In: Mathematical Annals. 83, 1921, §3, sentence IV., P. 36.
^ Emmy Noether: Ideal theory in ring areas. In: Mathematical Annals. 83, 1921, §4, sentence VII., P. 41.
^ Emmy Noether: Ideal theory in ring areas. In: Mathematical Annals. 83, 1921, §7, sentence XIII., P. 50.
↑ Ernst Kunz: Introduction to Algebraic Geometry. Vieweg, Braunschweig / Wiesbaden 1997, sentence C.30., P. 235.
↑ Ernst Kunz: Introduction to Algebraic Geometry. Vieweg, Braunschweig / Wiesbaden 1997, sentence C.32., P. 235
^ David Eisenbud: Commutative Algebra with a View Toward Algebraic Geometry. Springer, 1995, Proof of Theorem 3.10., P. 96.
^ Emmy Noether: Ideal theory in ring areas. In: Mathematical Annals. 83, 1921, §9, p. 56.
↑ Ladislas Fuchs: On Primal Ideals. In: Proceedings of the American Mathematical Society. Vol. 1, No. 1 (Feb., 1950), p. 3.

literature

Emmy Noether : Ideal theory in ring areas. In: Mathematical Annals . 83, 1921, pp. 24-66, GDZ
Michael Francis Atiyah , Ian Grant Macdonald : Introduction to Commutative Algebra. Addison-Wesley, Reading Mass. 1969, Chapter 7

[1] Michael Francis Atiyah, Ian Grant Macdonald: Introduction to Commutative Algebra. Addison-Wesley, Reading Mass. 1969, Proposition 1.11., P. 8

[2] Michael Francis Atiyah, Ian Grant Macdonald: Introduction to Commutative Algebra. Addison-Wesley, Reading Mass. 1969, Theorem 9.3., P. 95.

[3] Esra Miller, Bernd Sturmfels: Combinatorial Commutative Algebra. Springer, 2006, 5.16 f., 11.41.

[4] Emmy Noether: Ideal theory in ring areas. In: Mathematical Annals. 83, 1921, §4, sentence VI., P. 39.

[5] Michael Francis Atiyah, Ian Grant Macdonald: Introduction to Commutative Algebra. Addison-Wesley, Reading Mass. 1969, Lemma 7.12., P. 83.

[6] Is the radical of an irreducible ideal irreducible? mathoverflow. Retrieved December 8, 2019.

[7] Ladislas Fuchs: A condition under which an irreducible ideal is primary. In: The Quarterly Journal of Mathematics. Oxford Ser. 19 (1948), p. 235, Theorem 1

[8] Michael Francis Atiyah, Ian Grant Macdonald: Introduction to Commutative Algebra. Addison-Wesley, Reading Mass. 1969, Exercises 3., p. 84.

[9] Wolfgang Gröbner: About irreducible ideals in commutative rings. In: Mathematical Annals. 110, 1935, sentence 4a. P. 205, GDZ

[10] Emmy Noether: Ideal theory in ring areas. In: Mathematical Annals. 83, 1921, §2, sentence II., P. 33.

[11] Michael Francis Atiyah, Ian Grant Macdonald: Introduction to Commutative Algebra. Addison-Wesley, Reading Mass. 1969, Lemma 7.11., P. 83.

[12] Emmy Noether: Ideal theory in ring areas. In: Mathematical Annals. 83, 1921, §3, sentence IV., P. 36.

[13] Emmy Noether: Ideal theory in ring areas. In: Mathematical Annals. 83, 1921, §4, sentence VII., P. 41.

[14] Emmy Noether: Ideal theory in ring areas. In: Mathematical Annals. 83, 1921, §7, sentence XIII., P. 50.

[15] Ernst Kunz: Introduction to Algebraic Geometry. Vieweg, Braunschweig / Wiesbaden 1997, sentence C.30., P. 235.

[16] Ernst Kunz: Introduction to Algebraic Geometry. Vieweg, Braunschweig / Wiesbaden 1997, sentence C.32., P. 235

[17] David Eisenbud: Commutative Algebra with a View Toward Algebraic Geometry. Springer, 1995, Proof of Theorem 3.10., P. 96.

[18] Emmy Noether: Ideal theory in ring areas. In: Mathematical Annals. 83, 1921, §9, p. 56.

[19] Ladislas Fuchs: On Primal Ideals. In: Proceedings of the American Mathematical Society. Vol. 1, No. 1 (Feb., 1950), p. 3.