Volume 11, 1999

University of Bialystok

Copyright (c) 1999 Association of Mizar Users

**Christoph Schwarzweller**- University of T\"ubingen

- In this article we introduce the ring of Integers, Euclidean rings and Integers modulo $p$. In particular we prove that the Ring of Integers is an Euclidean ring and that the Integers modulo $p$ constitutes a field if and only if $p$ is a prime.

- The Ring of Integers
- Euclidean Rings
- Some Theorems about Div and Mod
- Modulo Integers

- [1]
Grzegorz Bancerek.
The fundamental properties of natural numbers.
*Journal of Formalized Mathematics*, 1, 1989. - [2]
Grzegorz Bancerek.
Sequences of ordinal numbers.
*Journal of Formalized Mathematics*, 1, 1989. - [3]
Grzegorz Bancerek and Andrzej Trybulec.
Miscellaneous facts about functions.
*Journal of Formalized Mathematics*, 8, 1996. - [4]
Czeslaw Bylinski.
Binary operations.
*Journal of Formalized Mathematics*, 1, 1989. - [5]
Czeslaw Bylinski.
Functions and their basic properties.
*Journal of Formalized Mathematics*, 1, 1989. - [6]
Czeslaw Bylinski.
Functions from a set to a set.
*Journal of Formalized Mathematics*, 1, 1989. - [7]
Marek Chmur.
The lattice of natural numbers and the sublattice of it. The set of prime numbers.
*Journal of Formalized Mathematics*, 3, 1991. - [8]
Agata Darmochwal.
The Euclidean space.
*Journal of Formalized Mathematics*, 3, 1991. - [9]
Krzysztof Hryniewiecki.
Basic properties of real numbers.
*Journal of Formalized Mathematics*, 1, 1989. - [10]
Eugeniusz Kusak, Wojciech Leonczuk, and Michal Muzalewski.
Abelian groups, fields and vector spaces.
*Journal of Formalized Mathematics*, 1, 1989. - [11]
Rafal Kwiatek and Grzegorz Zwara.
The divisibility of integers and integer relatively primes.
*Journal of Formalized Mathematics*, 2, 1990. - [12]
Michal Muzalewski.
Construction of rings and left-, right-, and bi-modules over a ring.
*Journal of Formalized Mathematics*, 2, 1990. - [13]
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. - [14]
Dariusz Surowik.
Cyclic groups and some of their properties --- part I.
*Journal of Formalized Mathematics*, 3, 1991. - [15]
Andrzej Trybulec.
Tarski Grothendieck set theory.
*Journal of Formalized Mathematics*, Axiomatics, 1989. - [16]
Andrzej Trybulec.
Subsets of real numbers.
*Journal of Formalized Mathematics*, Addenda, 2003. - [17]
Michal J. Trybulec.
Integers.
*Journal of Formalized Mathematics*, 2, 1990. - [18]
Wojciech A. Trybulec.
Vectors in real linear space.
*Journal of Formalized Mathematics*, 1, 1989. - [19]
Wojciech A. Trybulec.
Groups.
*Journal of Formalized Mathematics*, 2, 1990. - [20]
Zinaida Trybulec.
Properties of subsets.
*Journal of Formalized Mathematics*, 1, 1989. - [21]
Edmund Woronowicz.
Relations and their basic properties.
*Journal of Formalized Mathematics*, 1, 1989.

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