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
@article{bwmeta1.element.doi-10_2478_forma-2014-0029,
author = {Artur Korni\l owicz and Christoph Schwarzweller},
title = {The First Isomorphism Theorem and Other Properties of Rings},
journal = {Formalized Mathematics},
volume = {22},
year = {2014},
pages = {291-301},
zbl = {1316.13003},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.doi-10_2478_forma-2014-0029}
}
Artur Korniłowicz; Christoph Schwarzweller. The First Isomorphism Theorem and Other Properties of Rings. Formalized Mathematics, Tome 22 (2014) pp. 291-301. http://gdmltest.u-ga.fr/item/bwmeta1.element.doi-10_2478_forma-2014-0029/
[1] Jonathan Backer, Piotr Rudnicki, and Christoph Schwarzweller. Ring ideals. Formalized Mathematics, 9(3):565-582, 2001.
[2] Grzegorz Bancerek. Cardinal numbers. Formalized Mathematics, 1(2):377-382, 1990.
[3] Grzegorz Bancerek. The fundamental properties of natural numbers. Formalized Mathematics, 1(1):41-46, 1990. | Zbl 06213858
[4] Grzegorz Bancerek. The ordinal numbers. Formalized Mathematics, 1(1):91-96, 1990.
[5] Grzegorz Bancerek and Krzysztof Hryniewiecki. Segments of natural numbers and finite sequences. Formalized Mathematics, 1(1):107-114, 1990.
[6] Józef Białas. Group and field definitions. Formalized Mathematics, 1(3):433-439, 1990.
[7] Czesław Bylinski. Functions and their basic properties. Formalized Mathematics, 1(1): 55-65, 1990.
[8] Czesław Bylinski. Functions from a set to a set. Formalized Mathematics, 1(1):153-164, 1990.
[9] Czesław Bylinski. Partial functions. Formalized Mathematics, 1(2):357-367, 1990.
[10] Czesław Bylinski. Some basic properties of sets. Formalized Mathematics, 1(1):47-53, 1990.
[11] Agata Darmochwał. Finite sets. Formalized Mathematics, 1(1):165-167, 1990.
[12] Nathan Jacobson. Basic Algebra I. 2nd edition. Dover Publications Inc., 2009. | Zbl 0284.16001
[13] Andrzej Kondracki. Basic properties of rational numbers. Formalized Mathematics, 1(5): 841-845, 1990.
[14] Artur Korniłowicz. Quotient rings. Formalized Mathematics, 13(4):573-576, 2005.
[15] Jarosław Kotowicz. Quotient vector spaces and functionals. Formalized Mathematics, 11 (1):59-68, 2003.
[16] Eugeniusz Kusak, Wojciech Leonczuk, and Michał Muzalewski. Abelian groups, fields and vector spaces. Formalized Mathematics, 1(2):335-342, 1990.
[17] Heinz L¨uneburg. Die grundlegenden Strukturen der Algebra (in German). Oldenbourg Wisenschaftsverlag, 1999.
[18] Robert Milewski. The ring of polynomials. Formalized Mathematics, 9(2):339-346, 2001.
[19] Michał Muzalewski. Opposite rings, modules and their morphisms. Formalized Mathematics, 3(1):57-65, 1992.
[20] Michał Muzalewski. Category of rings. Formalized Mathematics, 2(5):643-648, 1991.
[21] Michał Muzalewski. Construction of rings and left-, right-, and bi-modules over a ring. Formalized Mathematics, 2(1):3-11, 1991.
[22] Michał Muzalewski and Wojciech Skaba. From loops to Abelian multiplicative groups with zero. Formalized Mathematics, 1(5):833-840, 1990.
[23] Beata Padlewska. Families of sets. Formalized Mathematics, 1(1):147-152, 1990.
[24] Konrad Raczkowski and Paweł Sadowski. Equivalence relations and classes of abstraction. Formalized Mathematics, 1(3):441-444, 1990.
[25] Piotr Rudnicki and Andrzej Trybulec. Multivariate polynomials with arbitrary number of variables. Formalized Mathematics, 9(1):95-110, 2001.
[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] Christoph Schwarzweller. The ring of integers, Euclidean rings and modulo integers. Formalized Mathematics, 8(1):29-34, 1999.
[28] Christoph Schwarzweller. The field of quotients over an integral domain. Formalized Mathematics, 7(1):69-79, 1998.
[29] Christoph Schwarzweller. Introduction to rational functions. Formalized Mathematics, 20 (2):181-191, 2012. doi:10.2478/v10037-012-0021-1.[Crossref] | Zbl 1285.26027
[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] 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] Andrzej Trybulec. Domains and their Cartesian products. Formalized Mathematics, 1(1): 115-122, 1990.
[33] Andrzej Trybulec. Binary operations applied to functions. Formalized Mathematics, 1 (2):329-334, 1990.
[34] Andrzej Trybulec. On the sets inhabited by numbers. Formalized Mathematics, 11(4): 341-347, 2003.
[35] Michał J. Trybulec. Integers. Formalized Mathematics, 1(3):501-505, 1990.
[36] Wojciech A. Trybulec. Groups. Formalized Mathematics, 1(5):821-827, 1990.
[37] Wojciech A. Trybulec. Lattice of subgroups of a group. Frattini subgroup. Formalized Mathematics, 2(1):41-47, 1991.
[38] Wojciech A. Trybulec. Vectors in real linear space. Formalized Mathematics, 1(2):291-296, 1990.
[39] Wojciech A. Trybulec and Michał J. Trybulec. Homomorphisms and isomorphisms of groups. Quotient group. Formalized Mathematics, 2(4):573-578, 1991.
[40] Zinaida Trybulec. Properties of subsets. Formalized Mathematics, 1(1):67-71, 1990.
[41] B.L. van der Waerden. Algebra I. 4th edition. Springer, 2003.
[42] Edmund Woronowicz. Relations and their basic properties. Formalized Mathematics, 1 (1):73-83, 1990.
[43] Edmund Woronowicz. Relations defined on sets. Formalized Mathematics, 1(1):181-186, 1990.