The First Isomorphism Theorem and Other Properties of Rings

Artur Korniłowicz; Christoph Schwarzweller

Formalized Mathematics (2014)

  • Volume: 22, Issue: 4, page 291-301
  • ISSN: 1426-2630

Abstract

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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

How to cite

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Artur Korniłowicz, and Christoph Schwarzweller. "The First Isomorphism Theorem and Other Properties of Rings." Formalized Mathematics 22.4 (2014): 291-301. <http://eudml.org/doc/270996>.

@article{ArturKorniłowicz2014,
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},
author = {Artur Korniłowicz, Christoph Schwarzweller},
journal = {Formalized Mathematics},
keywords = {commutative algebra; ring theory; first isomorphism theorem},
language = {eng},
number = {4},
pages = {291-301},
title = {The First Isomorphism Theorem and Other Properties of Rings},
url = {http://eudml.org/doc/270996},
volume = {22},
year = {2014},
}

TY - JOUR
AU - Artur Korniłowicz
AU - Christoph Schwarzweller
TI - The First Isomorphism Theorem and Other Properties of Rings
JO - Formalized Mathematics
PY - 2014
VL - 22
IS - 4
SP - 291
EP - 301
AB - 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
LA - eng
KW - commutative algebra; ring theory; first isomorphism theorem
UR - http://eudml.org/doc/270996
ER -

References

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