36,617 research outputs found
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 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.Korniłowicz Artur - Institute of Informatics University of Białystok Sosnowa 64, 15-887 Białystok PolandSchwarzweller Christoph - Institute of Computer Science University of Gdansk Wita Stwosza 57, 80-952 Gdansk PolandJonathan Backer, Piotr Rudnicki, and Christoph Schwarzweller. Ring ideals. Formalized Mathematics, 9(3):565-582, 2001.Grzegorz Bancerek. Cardinal numbers. Formalized Mathematics, 1(2):377-382, 1990.Grzegorz Bancerek. The fundamental properties of natural numbers. Formalized Mathematics, 1(1):41-46, 1990.Grzegorz Bancerek. The ordinal numbers. Formalized Mathematics, 1(1):91-96, 1990.Grzegorz Bancerek and Krzysztof Hryniewiecki. Segments of natural numbers and finite sequences. Formalized Mathematics, 1(1):107-114, 1990.Józef Białas. Group and field definitions. Formalized Mathematics, 1(3):433-439, 1990.Czesław Bylinski. Functions and their basic properties. Formalized Mathematics, 1(1): 55-65, 1990.Czesław Bylinski. Functions from a set to a set. Formalized Mathematics, 1(1):153-164, 1990.Czesław Bylinski. Partial functions. Formalized Mathematics, 1(2):357-367, 1990.Czesław Bylinski. Some basic properties of sets. Formalized Mathematics, 1(1):47-53, 1990.Agata Darmochwał. Finite sets. Formalized Mathematics, 1(1):165-167, 1990.Nathan Jacobson. Basic Algebra I. 2nd edition. Dover Publications Inc., 2009.Andrzej Kondracki. Basic properties of rational numbers. Formalized Mathematics, 1(5): 841-845, 1990.Artur Korniłowicz. Quotient rings. Formalized Mathematics, 13(4):573-576, 2005.Jarosław Kotowicz. Quotient vector spaces and functionals. Formalized Mathematics, 11 (1):59-68, 2003.Eugeniusz Kusak, Wojciech Leonczuk, and Michał Muzalewski. Abelian groups, fields and vector spaces. Formalized Mathematics, 1(2):335-342, 1990.Heinz L¨uneburg. Die grundlegenden Strukturen der Algebra (in German). Oldenbourg Wisenschaftsverlag, 1999.Robert Milewski. The ring of polynomials. Formalized Mathematics, 9(2):339-346, 2001.Michał Muzalewski. Opposite rings, modules and their morphisms. Formalized Mathematics, 3(1):57-65, 1992.Michał Muzalewski. Category of rings. Formalized Mathematics, 2(5):643-648, 1991.Michał Muzalewski. Construction of rings and left-, right-, and bi-modules over a ring. Formalized Mathematics, 2(1):3-11, 1991.Michał Muzalewski and Wojciech Skaba. From loops to Abelian multiplicative groups with zero. Formalized Mathematics, 1(5):833-840, 1990.Beata Padlewska. Families of sets. Formalized Mathematics, 1(1):147-152, 1990.Konrad Raczkowski and Paweł Sadowski. Equivalence relations and classes of abstraction. Formalized Mathematics, 1(3):441-444, 1990.Piotr Rudnicki and Andrzej Trybulec. Multivariate polynomials with arbitrary number of variables. Formalized Mathematics, 9(1):95-110, 2001.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.Christoph Schwarzweller. The ring of integers, Euclidean rings and modulo integers. Formalized Mathematics, 8(1):29-34, 1999.Christoph Schwarzweller. The field of quotients over an integral domain. Formalized Mathematics, 7(1):69-79, 1998.Christoph Schwarzweller. Introduction to rational functions. Formalized Mathematics, 20 (2):181-191, 2012. doi:10.2478/v10037-012-0021-1.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.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.Andrzej Trybulec. Domains and their Cartesian products. Formalized Mathematics, 1(1): 115-122, 1990.Andrzej Trybulec. Binary operations applied to functions. Formalized Mathematics, 1 (2):329-334, 1990.Andrzej Trybulec. On the sets inhabited by numbers. Formalized Mathematics, 11(4): 341-347, 2003.Michał J. Trybulec. Integers. Formalized Mathematics, 1(3):501-505, 1990.Wojciech A. Trybulec. Groups. Formalized Mathematics, 1(5):821-827, 1990.Wojciech A. Trybulec. Lattice of subgroups of a group. Frattini subgroup. Formalized Mathematics, 2(1):41-47, 1991.Wojciech A. Trybulec. Vectors in real linear space. Formalized Mathematics, 1(2):291-296, 1990.Wojciech A. Trybulec and Michał J. Trybulec. Homomorphisms and isomorphisms of groups. Quotient group. Formalized Mathematics, 2(4):573-578, 1991.Zinaida Trybulec. Properties of subsets. Formalized Mathematics, 1(1):67-71, 1990.B.L. van der Waerden. Algebra I. 4th edition. Springer, 2003.Edmund Woronowicz. Relations and their basic properties. Formalized Mathematics, 1 (1):73-83, 1990.Edmund Woronowicz. Relations defined on sets. Formalized Mathematics, 1(1):181-186, 1990
Locally complete intersection homomorphisms and a conjecture of Quillen on the vanishing of cotangent homology
Classical definitions of locally complete intersection (l.c.i.) homomorphisms
of commutative rings are limited to maps that are essentially of finite type,
or flat. The concept introduced in this paper is meaningful for homomorphisms
phi : R \longrightarrow S of commutative noetherian rings. It is defined in
terms of the structure of phi in a formal neighborhood of each point of Spec S.
We characterize the l.c.i. property by different conditions on the vanishing of
the Andr\'e-Quillen homology of the R-algebra S. One of these descriptions
establishes a very general form of a conjecture of Quillen that was open even
for homomorphisms of finite type: If S has a finite resolution by flat
R-modules and the cotangent complex \cot SR is quasi-isomorphic to a bounded
complex of flat S-modules, then phi is l.c.i. The proof uses a mixture of
methods from commutative algebra, differential graded homological algebra, and
homotopy theory. The l.c.i. property is shown to be stable under a variety of
operations, including composition, decomposition, flat base change,
localization, and completion. The present framework allows for the results to
be stated in proper generality; many of them are new even with classical
assumptions. For instance, the stability of l.c.i. homomorphisms under
decomposition settles an open case in Fulton's treatment of orientations of
morphisms of schemes.Comment: 33 pages, published versio
Gorenstein algebras and Hochschild cohomology
For homomorphism K-->S of commutative rings, where K is Gorenstein and S is
essentially of finite type and flat as a K-module, the property that all
non-trivial fiber rings of K-->S are Gorenstein is characterized in terms of
properties of the cohomology modules Ext_n^{S\otimes_KS}S{S\otimes_KS}.Comment: This is the published version, except for updates to references and
bibliography. Sections 3, 4 and 8 have been removed from the preceding
version, arXiv:0704.3761v2. Substantial generalizations of results in those
sections are proved in our paper with Joseph Lipman and Suresh Nayak,
arXiv:0904.400
Leavitt path algebras: the first decade
The algebraic structures known as {\it Leavitt path algebras} were initially
developed in 2004 by Ara, Moreno and Pardo, and almost simultaneously (using a
different approach) by the author and Aranda Pino.
During the intervening decade, these algebras have attracted significant
interest and attention, not only from ring theorists, but from analysts working
in C-algebras, group theorists, and symbolic dynamicists as well. The goal
of this article is threefold: to introduce the notion of Leavitt path algebras
to the general mathematical community; to present some of the important results
in the subject; and to describe some of the field's currently unresolved
questions.Comment: 53 pages. To appear, Bulletin of Mathematical Sciences. (page
numbering in arXiv version will differ from page numbering in BMS published
version; numbering of Theorems, etc ... will be the same in both versions
Equivariant cobordism of schemes
We study the equivariant cobordism theory of schemes for action of linear
algebraic groups. We compare the equivariant cobordism theory for the action of
a linear algebraic groups with similar groups for the action of tori and deduce
some consequences for the cycle class map of the classifying space of an
algebraic groups.Comment: This revised version supercedes arxiv:1006:317
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