# The First Isomorphism Theorem and Other Properties of Rings

Formalized Mathematics (2014)

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

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## 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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1. [1] Jonathan Backer, Piotr Rudnicki, and Christoph Schwarzweller. Ring ideals. Formalized Mathematics, 9(3):565-582, 2001.
2. [2] Grzegorz Bancerek. Cardinal numbers. Formalized Mathematics, 1(2):377-382, 1990.
3. [3] Grzegorz Bancerek. The fundamental properties of natural numbers. Formalized Mathematics, 1(1):41-46, 1990. Zbl06213858
4. [4] Grzegorz Bancerek. The ordinal numbers. Formalized Mathematics, 1(1):91-96, 1990.
5. [5] Grzegorz Bancerek and Krzysztof Hryniewiecki. Segments of natural numbers and finite sequences. Formalized Mathematics, 1(1):107-114, 1990.
6. [6] Józef Białas. Group and field definitions. Formalized Mathematics, 1(3):433-439, 1990.
7. [7] Czesław Bylinski. Functions and their basic properties. Formalized Mathematics, 1(1): 55-65, 1990.
8. [8] Czesław Bylinski. Functions from a set to a set. Formalized Mathematics, 1(1):153-164, 1990.
9. [9] Czesław Bylinski. Partial functions. Formalized Mathematics, 1(2):357-367, 1990.
10. [10] Czesław Bylinski. Some basic properties of sets. Formalized Mathematics, 1(1):47-53, 1990.
11. [11] Agata Darmochwał. Finite sets. Formalized Mathematics, 1(1):165-167, 1990.
12. [12] Nathan Jacobson. Basic Algebra I. 2nd edition. Dover Publications Inc., 2009. Zbl0284.16001
13. [13] Andrzej Kondracki. Basic properties of rational numbers. Formalized Mathematics, 1(5): 841-845, 1990.
14. [14] Artur Korniłowicz. Quotient rings. Formalized Mathematics, 13(4):573-576, 2005.
15. [15] Jarosław Kotowicz. Quotient vector spaces and functionals. Formalized Mathematics, 11 (1):59-68, 2003.
16. [16] Eugeniusz Kusak, Wojciech Leonczuk, and Michał Muzalewski. Abelian groups, fields and vector spaces. Formalized Mathematics, 1(2):335-342, 1990.
17. [17] Heinz L¨uneburg. Die grundlegenden Strukturen der Algebra (in German). Oldenbourg Wisenschaftsverlag, 1999.
18. [18] Robert Milewski. The ring of polynomials. Formalized Mathematics, 9(2):339-346, 2001.
19. [19] Michał Muzalewski. Opposite rings, modules and their morphisms. Formalized Mathematics, 3(1):57-65, 1992.
20. [20] Michał Muzalewski. Category of rings. Formalized Mathematics, 2(5):643-648, 1991.
21. [21] Michał Muzalewski. Construction of rings and left-, right-, and bi-modules over a ring. Formalized Mathematics, 2(1):3-11, 1991.
22. [22] Michał Muzalewski and Wojciech Skaba. From loops to Abelian multiplicative groups with zero. Formalized Mathematics, 1(5):833-840, 1990.
23. [23] Beata Padlewska. Families of sets. Formalized Mathematics, 1(1):147-152, 1990.
24. [24] Konrad Raczkowski and Paweł Sadowski. Equivalence relations and classes of abstraction. Formalized Mathematics, 1(3):441-444, 1990.
25. [25] Piotr Rudnicki and Andrzej Trybulec. Multivariate polynomials with arbitrary number of variables. Formalized Mathematics, 9(1):95-110, 2001.
26. [26] Christoph Schwarzweller. The correctness of the generic algorithms of Brown and Henrici concerning addition and multiplication in fraction fields. Formalized Mathematics, 6(3): 381-388, 1997.
27. [27] Christoph Schwarzweller. The ring of integers, Euclidean rings and modulo integers. Formalized Mathematics, 8(1):29-34, 1999.
28. [28] Christoph Schwarzweller. The field of quotients over an integral domain. Formalized Mathematics, 7(1):69-79, 1998.
29. [29] Christoph Schwarzweller. Introduction to rational functions. Formalized Mathematics, 20 (2):181-191, 2012. doi:10.2478/v10037-012-0021-1.[Crossref] Zbl1285.26027
30. [30] Christoph Schwarzweller and Agnieszka Rowinska-Schwarzweller. Schur’s theorem on the stability of networks. Formalized Mathematics, 14(4):135-142, 2006. doi:10.2478/v10037-006-0017-9.[Crossref]
31. [31] Yasunari Shidama, Hikofumi Suzuki, and Noboru Endou. Banach algebra of bounded functionals. Formalized Mathematics, 16(2):115-122, 2008. doi:10.2478/v10037-008-0017- z.[Crossref]
32. [32] Andrzej Trybulec. Domains and their Cartesian products. Formalized Mathematics, 1(1): 115-122, 1990.
33. [33] Andrzej Trybulec. Binary operations applied to functions. Formalized Mathematics, 1 (2):329-334, 1990.
34. [34] Andrzej Trybulec. On the sets inhabited by numbers. Formalized Mathematics, 11(4): 341-347, 2003.
35. [35] Michał J. Trybulec. Integers. Formalized Mathematics, 1(3):501-505, 1990.
36. [36] Wojciech A. Trybulec. Groups. Formalized Mathematics, 1(5):821-827, 1990.
37. [37] Wojciech A. Trybulec. Lattice of subgroups of a group. Frattini subgroup. Formalized Mathematics, 2(1):41-47, 1991.
38. [38] Wojciech A. Trybulec. Vectors in real linear space. Formalized Mathematics, 1(2):291-296, 1990.
39. [39] Wojciech A. Trybulec and Michał J. Trybulec. Homomorphisms and isomorphisms of groups. Quotient group. Formalized Mathematics, 2(4):573-578, 1991.
40. [40] Zinaida Trybulec. Properties of subsets. Formalized Mathematics, 1(1):67-71, 1990.
41. [41] B.L. van der Waerden. Algebra I. 4th edition. Springer, 2003.
42. [42] Edmund Woronowicz. Relations and their basic properties. Formalized Mathematics, 1 (1):73-83, 1990.
43. [43] Edmund Woronowicz. Relations defined on sets. Formalized Mathematics, 1(1):181-186, 1990.

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