<abstract xmlns="http://www.w3.org/1999/xhtml"><p> Different properties of rings and fields are discussed [12], [41] and [17]. We introduce ring homomorphisms, their kernels and images, and prove the First Isomorphism Theorem, namely that for a homomorphism f : R → S we have R/ker(f) ≅ Im(f). Then we define prime and irreducible elements and show that every principal ideal domain is factorial. Finally we show that polynomial rings over fields are Euclidean and hence also factorial </p></abstract>

Different properties of rings and fields are discussed [12], [41] and [17]. We introduce ring homomorphisms, their kernels and images, and prove the First Isomorphism Theorem, namely that for a homomorphism f : R → S we have R/ker(f) ≅ Im(f). Then we define prime and irreducible elements and show that every principal ideal domain is factorial. Finally we show that polynomial rings over fields are Euclidean and hence also factorial

The First Isomorphism Theorem and Other Properties of Rings

Institute of Informatics University of Białystok Sosnowa 64, 15-887 Białystok Poland

Institute of Computer Science University of Gdansk Wita Stwosza 57, 80-952 Gdansk Poland

Formalized Mathematics

This work is licensed under the Creative Commons Attribution-ShareAlike 3.0 License.

{"article-title":"The First Isomorphism Theorem and Other Properties of Rings"}

Different properties of rings and fields are discussed [12], [41] and [17]. We introduce ring homomorphisms, their kernels and images, and prove the First...

The First Isomorphism Theorem and Other Properties of Rings

Published Online: Dec 31, 2014

Page range: 291 - 301

Received: Nov 29, 2014

DOI: https://doi.org/10.2478/forma-2014-0029

Keywords
commutative algebra, ring theory, first isomorphism theorem

© by Artur Korniłowicz

This work is licensed under the Creative Commons Attribution-ShareAlike 3.0 License.

The First Isomorphism Theorem and Other Properties of Rings

Published Online: Dec 31, 2014

Page range: 291 - 301

Received: Nov 29, 2014

DOI: https://doi.org/10.2478/forma-2014-0029

Keywordscommutative algebra, ring theory, first isomorphism theorem

© by Artur Korniłowicz

This work is licensed under the Creative Commons Attribution-ShareAlike 3.0 License.

Keywords
commutative algebra, ring theory, first isomorphism theorem