Journal of Formalized Mathematics
Volume 12, 2000
University of Bialystok
Copyright (c) 2000 Association of Mizar Users

## Ring Ideals

Jonathan Backer
University of Alberta, Edmonton
Partially supported by NSERC grant OGP9207.
Piotr Rudnicki
University of Alberta, Edmonton
Partially supported by NSERC grant OGP9207.
Christoph Schwarzweller
University of T\"ubingen
Partially supported by CALCULEMUS grant HPRN-CT-2000-00102.

### Summary.

We introduce the basic notions of ideal theory in rings. This includes left and right ideals, (finitely) generated ideals and some operations on ideals such as the addition of ideals and the radical of an ideal. In addition we introduce linear combinations to formalize the well-known characterization of generated ideals. Principal ideal domains and Noetherian rings are defined. The latter development follows [4], pages 144-145.

#### MML Identifier: IDEAL_1

The terminology and notation used in this paper have been introduced in the following articles [23] [9] [29] [10] [6] [2] [11] [19] [16] [25] [1] [30] [26] [8] [7] [13] [15] [21] [24] [22] [28] [17] [14] [27] [12] [3] [5] [18] [20]

#### Contents (PDF format)

1. Preliminaries
2. Ideals
3. Linear Combinations
4. Generated Ideals
5. Some Operations on Ideals
6. Noetherian Rings and PIDs

#### Bibliography

[1] Grzegorz Bancerek. The fundamental properties of natural numbers. Journal of Formalized Mathematics, 1, 1989.
[2] Grzegorz Bancerek and Krzysztof Hryniewiecki. Segments of natural numbers and finite sequences. Journal of Formalized Mathematics, 1, 1989.
[3] Grzegorz Bancerek and Andrzej Trybulec. Miscellaneous facts about functions. Journal of Formalized Mathematics, 8, 1996.
[4] Thomas Becker and Volker Weispfenning. \em Gr\"obner Bases: A Computational Approach to Commutative Algebra. Springer-Verlag, New York, Berlin, 1993.
[5] Jozef Bialas. Group and field definitions. Journal of Formalized Mathematics, 1, 1989.
[6] Czeslaw Bylinski. Functions and their basic properties. Journal of Formalized Mathematics, 1, 1989.
[7] Czeslaw Bylinski. Functions from a set to a set. Journal of Formalized Mathematics, 1, 1989.
[8] Czeslaw Bylinski. Partial functions. Journal of Formalized Mathematics, 1, 1989.
[9] Czeslaw Bylinski. Some basic properties of sets. Journal of Formalized Mathematics, 1, 1989.
[10] Agata Darmochwal. Finite sets. Journal of Formalized Mathematics, 1, 1989.
[11] Eugeniusz Kusak, Wojciech Leonczuk, and Michal Muzalewski. Abelian groups, fields and vector spaces. Journal of Formalized Mathematics, 1, 1989.
[12] Michal Muzalewski. Construction of rings and left-, right-, and bi-modules over a ring. Journal of Formalized Mathematics, 2, 1990.
[13] Michal Muzalewski and Wojciech Skaba. From loops to abelian multiplicative groups with zero. Journal of Formalized Mathematics, 2, 1990.
[14] Beata Padlewska. Families of sets. Journal of Formalized Mathematics, 1, 1989.
[15] Beata Padlewska and Agata Darmochwal. Topological spaces and continuous functions. Journal of Formalized Mathematics, 1, 1989.
[16] Jan Popiolek. Real normed space. Journal of Formalized Mathematics, 2, 1990.
[17] Piotr Rudnicki and Andrzej Trybulec. Multivariate polynomials with arbitrary number of variables. Journal of Formalized Mathematics, 11, 1999.
[18] Christoph Schwarzweller. The correctness of the generic algorithms of Brown and Henrici concerning addition and multiplication in fraction fields. Journal of Formalized Mathematics, 9, 1997.
[19] Christoph Schwarzweller. The ring of integers, euclidean rings and modulo integers. Journal of Formalized Mathematics, 11, 1999.
[20] Christoph Schwarzweller. The binomial theorem for algebraic structures. Journal of Formalized Mathematics, 12, 2000.
[21] Dariusz Surowik. Cyclic groups and some of their properties --- part I. Journal of Formalized Mathematics, 3, 1991.
[22] Andrzej Trybulec. Domains and their Cartesian products. Journal of Formalized Mathematics, 1, 1989.
[23] Andrzej Trybulec. Tarski Grothendieck set theory. Journal of Formalized Mathematics, Axiomatics, 1989.
[24] Andrzej Trybulec. Tuples, projections and Cartesian products. Journal of Formalized Mathematics, 1, 1989.
[25] Andrzej Trybulec. Subsets of real numbers. Journal of Formalized Mathematics, Addenda, 2003.
[26] Wojciech A. Trybulec. Vectors in real linear space. Journal of Formalized Mathematics, 1, 1989.
[27] Wojciech A. Trybulec. Groups. Journal of Formalized Mathematics, 2, 1990.
[28] Wojciech A. Trybulec. Pigeon hole principle. Journal of Formalized Mathematics, 2, 1990.
[29] Zinaida Trybulec. Properties of subsets. Journal of Formalized Mathematics, 1, 1989.
[30] Edmund Woronowicz. Relations defined on sets. Journal of Formalized Mathematics, 1, 1989.

Received November 20, 2000