Uniquely Separable Extensions
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Ring Extension Problem, Shukla Cohomology and Ann-Category Theory the Relationship Between Ring Extension Problem in the General Case and Ann- Category Theory
RING EXTENSION PROBLEM, SHUKLA COHOMOLOGY AND ANN-CATEGORY THEORY Nguyen Tien Quang and Nguyen Thu Thuy November 4, 2018 Abstract. Every ring extension of A by R induces a pair of group ho- ∗ ∗ momorphisms L : R → EndZ(A)/L(A); R : R → EndZ(A)/R(A), preserving multiplication, satisfying some certain conditions. A such 4-tuple (R, A, L∗, R∗) is called a ring pre-extension. Each ring pre-extension induces a R-bimodule structure on bicenter KA of ring A, and induces an obstruction k, which is a 3-cocycle of Z-algebra R, with coefficients in R-bimodule KA in the sense of Shukla. Each obstruction k in this sense induces a structure of a regular Ann-category of type (R,KA). This result gives us the first application of Ann- category in extension problems of algebraic structures, as well as in cohomology theories. 1 Introduction Group extension problem has been presented with group of automorphisms Aut(G) and quotient group Aut(G)/In(G) by group of inner automorphisms. For ring extension problem, Mac Lane [1] has replaced the above groups by the ring of bimultiplications MA and the quotient ring PA of MA upon ring of in- ner bimultiplications. Besides, Maclane has replaced commutative, associative laws for addition in ring R by commutative-associative law (u + v) + (r + s)= arXiv:0706.0315v1 [math.CT] 3 Jun 2007 (u+r)+(v+s) and therefore proved that each obstruction of ring extension prob- lem is an element of 3-dimensional cohomology group in the sense of Maclane, and the number of solutions corresponds to 2-dimensional cohomology group of ring under a bijection. -
Isomorphic Groups and Group Rings
Pacific Journal of Mathematics ISOMORPHIC GROUPS AND GROUP RINGS DONALD STEVEN PASSMAN Vol. 15, No. 2 October 1965 PACIFIC JOURNAL OF MATHEMATICS Vol. 15, No. 2, 1965 ISOMORPHIC GROUPS AND GROUP RINGS D. S. PASSMAN Let © be a finite group, £ a commutative ring with one and S[@] the group ring of © over S. If ξ> is a group with © = £ then clearly S[(S] = S[£>] where the latter is an S-iso- morphism. We study here the converse question: For which groups © and rings S does £[©] ^ S[ξ>] imply that © is iso- morphic to £)? We consider first the case where S = K is a field. It is known that if © is abelian then Q[@] = Q[ξ>] implies that © = §> where Q is the field of rational numbers. We show here that this result does not extend to all groups ©. In fact by a simple counting argument we exhibit a large set of noniso- morphic p-groups with isomorphic group algebras over all noncharacteristic p fields. Thus for groups in general the only fields if interest are those whose characteristic divides the order of the group. We now let S = R be the ring of integers in some finite algebraic extension of the rationale. We show here that the group ring R[@>] determines the set of normal subgroups of © along with many of the natural operations defined on this set. For example, under the assumption that © is nilpotent, we show that given normal subgroups 3Dΐ and 9ΐ, the group ring determines the commutator subgroup (3JI, 91). Finally we consider several special cases. In particular we show that if © is nilpotent of class 2 then R[(g\ = β[§] implies © = €>. -
Section X.51. Separable Extensions
X.51 Separable Extensions 1 Section X.51. Separable Extensions Note. Let E be a finite field extension of F . Recall that [E : F ] is the degree of E as a vector space over F . For example, [Q(√2, √3) : Q] = 4 since a basis for Q(√2, √3) over Q is 1, √2, √3, √6 . Recall that the number of isomorphisms of { } E onto a subfield of F leaving F fixed is the index of E over F , denoted E : F . { } We are interested in when [E : F ]= E : F . When this equality holds (again, for { } finite extensions) E is called a separable extension of F . Definition 51.1. Let f(x) F [x]. An element α of F such that f(α) = 0 is a zero ∈ of f(x) of multiplicity ν if ν is the greatest integer such that (x α)ν is a factor of − f(x) in F [x]. Note. I find the following result unintuitive and surprising! The backbone of the (brief) proof is the Conjugation Isomorphism Theorem and the Isomorphism Extension Theorem. Theorem 51.2. Let f(x) be irreducible in F [x]. Then all zeros of f(x) in F have the same multiplicity. Note. The following follows from the Factor Theorem (Corollary 23.3) and Theo- rem 51.2. X.51 Separable Extensions 2 Corollary 51.3. If f(x) is irreducible in F [x], then f(x) has a factorization in F [x] of the form ν a Y(x αi) , i − where the αi are the distinct zeros of f(x) in F and a F . -
Math 210B. Inseparable Extensions Since the Theory of Non-Separable Algebraic Extensions Is Only Non-Trivial in Positive Charact
Math 210B. Inseparable extensions Since the theory of non-separable algebraic extensions is only non-trivial in positive characteristic, for this handout we shall assume all fields have positive characteristic p. 1. Separable and inseparable degree Let K=k be a finite extension, and k0=k the separable closure of k in K, so K=k0 is purely inseparable. 0 This yields two refinements of the field degree: the separable degree [K : k]s := [k : k] and the inseparable 0 degree [K : k]i := [K : k ] (so their product is [K : k], and [K : k]i is always a p-power). pn Example 1.1. Suppose K = k(a), and f 2 k[x] is the minimal polynomial of a. Then we have f = fsep(x ) pn where fsep 2 k[x] is separable irreducible over k, and a is a root of fsep (so the monic irreducible fsep is n n the minimal polynomial of ap over k). Thus, we get a tower K=k(ap )=k whose lower layer is separable and n upper layer is purely inseparable (as K = k(a)!). Hence, K=k(ap ) has no subextension that is a nontrivial n separable extension of k0 (why not?), so k0 = k(ap ), which is to say 0 pn [k(a): k]s = [k : k] = [k(a ): k] = deg fsep; 0 0 n [k(a): k]i = [K : k ] = [K : k]=[k : k] = (deg f)=(deg fsep) = p : If one tries to prove directly that the separable and inseparable degrees are multiplicative in towers just from the definitions, one runs into the problem that in general one cannot move all inseparability to the \bottom" of a finite extension (in contrast with the separability). -
Trivial Ring Extension of Suitable-Like Conditions and Some
An - Najah Univ. J. Res.(N. Sc.) Vol. 33(1), 2019 Trivial Ring Extension of Suitable-Like Conditions and some properties AmÌ'@ ªK.ð AîEAîD J.ð éJ¯A¾JÖÏ@ HA ®ÊjÊË éJ îE YJ.Ë@ éJ ®ÊmÌ'@ H@YK YÒJË@ Khalid Adarbeh* éK.P@Y« YËAg Department of Mathematics, Faculty of Sciences, An-Najah National University, Nablus, Palestine * Corresponding Author: [email protected] Received: (6/9/2018), Accepted: (6/12/2018) Abstract We investigate the transfer of the notion of suitable rings along with re- lated concepts, such as potent and semipotent rings, in the general context of the trivial ring extension, then we put these results in use to enrich the literature with new illustrative and counter examples subject to these ring- theoretic notions. Also we discuss some basic properties of the mentioned notions. Keywords: Trivial ring extension, idealization, clean ring, potent ring, semipotent ring, von Neumann regular ring, suitable rings. jÊÓ Ì « ªK. ð éJ¯A¾ÖÏ@ HA®Êm '@ Ðñê®Ó ÈA®JK@ éJ ÊÔ PñÖÏ@@Yë ú ¯ Ij.K Ì Ì áÓ éÓAªË@ éËAm '@ ú ¯ éK ñ®Ë@ éJ. ð éK ñ®Ë@ HA®Êm '@ ú ×ñê®Ò» é¢J.KQÖÏ@ Õæ ëA®ÖÏ@ YK ðQ K ú¯ èYK Ym.Ì'@ l.'AJJË@ ÐYjJ Õç' áÓð , éJ îE YJ.Ë@ éJ ®ÊmÌ'@ H@YK YÒJË@ ªK . ¯AJ K AÒ» . éðPYÖÏ@ ÕæëA®ÒÊË © m' éJ j ñ K éÊJÓAK . HA«ñJ .¢ÖÏ@ .Õæ ëA®ÖÏ@ èYêË éJ ®ÊmÌ'@ AmÌ'@ , é®J ¢ JË@ é®ÊmÌ'@ , éJ ËAJÖ ÏAK. ÉÒªË@ , éJ îE YJ.Ë@ éJ ®ÊmÌ'@ H@YK YÒJË@ : éJ kAJ®ÖÏ@ HAÒʾË@ . -
494 Lecture: Splitting Fields and the Primitive Element Theorem
494 Lecture: Splitting fields and the primitive element theorem Ben Gould March 30, 2018 1 Splitting Fields We saw previously that for any field F and (let's say irreducible) polynomial f 2 F [x], that there is some extension K=F such that f splits over K, i.e. all of the roots of f lie in K. We consider such extensions in depth now. Definition 1.1. For F a field and f 2 F [x], we say that an extension K=F is a splitting field for f provided that • f splits completely over K, i.e. f(x) = (x − a1) ··· (x − ar) for ai 2 K, and • K is generated by the roots of f: K = F (a1; :::; ar). The second statement implies that for each β 2 K there is a polynomial p 2 F [x] such that p(a1; :::; ar) = β; in general such a polynomial is not unique (since the ai are algebraic over F ). When F contains Q, it is clear that the splitting field for f is unique (up to isomorphism of fields). In higher characteristic, one needs to construct the splitting field abstractly. A similar uniqueness result can be obtained. Proposition 1.2. Three things about splitting fields. 1. If K=L=F is a tower of fields such that K is the splitting field for some f 2 F [x], K is also the splitting field of f when considered in K[x]. 2. Every polynomial over any field (!) admits a splitting field. 3. A splitting field is a finite extension of the base field, and every finite extension is contained in a splitting field. -
Commutative Algebra
Commutative Algebra Andrew Kobin Spring 2016 / 2019 Contents Contents Contents 1 Preliminaries 1 1.1 Radicals . .1 1.2 Nakayama's Lemma and Consequences . .4 1.3 Localization . .5 1.4 Transcendence Degree . 10 2 Integral Dependence 14 2.1 Integral Extensions of Rings . 14 2.2 Integrality and Field Extensions . 18 2.3 Integrality, Ideals and Localization . 21 2.4 Normalization . 28 2.5 Valuation Rings . 32 2.6 Dimension and Transcendence Degree . 33 3 Noetherian and Artinian Rings 37 3.1 Ascending and Descending Chains . 37 3.2 Composition Series . 40 3.3 Noetherian Rings . 42 3.4 Primary Decomposition . 46 3.5 Artinian Rings . 53 3.6 Associated Primes . 56 4 Discrete Valuations and Dedekind Domains 60 4.1 Discrete Valuation Rings . 60 4.2 Dedekind Domains . 64 4.3 Fractional and Invertible Ideals . 65 4.4 The Class Group . 70 4.5 Dedekind Domains in Extensions . 72 5 Completion and Filtration 76 5.1 Topological Abelian Groups and Completion . 76 5.2 Inverse Limits . 78 5.3 Topological Rings and Module Filtrations . 82 5.4 Graded Rings and Modules . 84 6 Dimension Theory 89 6.1 Hilbert Functions . 89 6.2 Local Noetherian Rings . 94 6.3 Complete Local Rings . 98 7 Singularities 106 7.1 Derived Functors . 106 7.2 Regular Sequences and the Koszul Complex . 109 7.3 Projective Dimension . 114 i Contents Contents 7.4 Depth and Cohen-Macauley Rings . 118 7.5 Gorenstein Rings . 127 8 Algebraic Geometry 133 8.1 Affine Algebraic Varieties . 133 8.2 Morphisms of Affine Varieties . 142 8.3 Sheaves of Functions . -
Proquest Dissertations
University of Alberta Descent Constructions for Central Extensions of Infinite Dimensional Lie Algebras by Jie Sun A thesis submitted to the Faculty of Graduate Studies and Research in partial fulfillment of the requirements for the degree of Doctor of Philosophy in Mathematics Department of Mathematical and Statistical Sciences Edmonton, Alberta Spring 2009 Library and Archives Bibliotheque et 1*1 Canada Archives Canada Published Heritage Direction du Branch Patrimoine de I'edition 395 Wellington Street 395, rue Wellington OttawaONK1A0N4 OttawaONK1A0N4 Canada Canada Your file Votre rSterence ISBN: 978-0-494-55620-7 Our file Notre rGterence ISBN: 978-0-494-55620-7 NOTICE: AVIS: The author has granted a non L'auteur a accorde une licence non exclusive exclusive license allowing Library and permettant a la Bibliotheque et Archives Archives Canada to reproduce, Canada de reproduire, publier, archiver, publish, archive, preserve, conserve, sauvegarder, conserver, transmettre au public communicate to the public by par telecommunication ou par I'lnternet, preter, telecommunication or on the Internet, distribuer et vendre des theses partout dans le loan, distribute and sell theses monde, a des fins commerciales ou autres, sur worldwide, for commercial or non support microforme, papier, electronique et/ou commercial purposes, in microform, autres formats. paper, electronic and/or any other formats. The author retains copyright L'auteur conserve la propriete du droit d'auteur ownership and moral rights in this et des droits moraux qui protege cette these. Ni thesis. Neither the thesis nor la these ni des extraits substantiels de celle-ci substantial extracts from it may be ne doivent etre im primes ou autrement printed or otherwise reproduced reproduits sans son autorisation. -
ON DENSITY of PRIMITIVE ELEMENTS for FIELD EXTENSIONS It Is Well Known That
ON DENSITY OF PRIMITIVE ELEMENTS FOR FIELD EXTENSIONS JOEL V. BRAWLEY AND SHUHONG GAO Abstract. This paper presents an explicit bound on the number of primitive ele- ments that are linear combinations of generators for field extensions. It is well known that every finite separable extension of an arbitrary field has a primitive element; that is, there exists a single element in the extension field which generates that field over the ground field. This is a fundamental theorem in algebra which is called the primitive element theorem in many textbooks, see for example [2, 5, 6], and it is a useful tool in practical computation of commutative algebra [4]. The existence proofs found in the literature make no attempt at estimating the density of primitive elements. The purpose of this paper is to give an explicit lower bound on the density of primitive elements that are linear combinations of generators. Our derivation uses a blend of Galois theory, basic linear algebra, and a simple form of the principle of inclusion-exclusion from elementary combinatorics. Additionally, for readers familiar with Grobner bases, we show by a geometric example, how to test a linear combination for primitivity without relying on the Galois groups used in deriving our bound. Let F be any field and K a finite algebraic extension of F. An element β ∈ K is called primitive for K over F if K = F(β). Suppose K is generated by α1,...,αn, that is, K = F(α1,...,αn). Consider elements of the form β = b1α1 + ···+bnαn where bi ∈ F,1≤i≤n. -
Notes on Commutative Algebra
Notes on Commutative Algebra Dan Segal February 2015 1 Preliminary definitions etc. Rings: commutative, with identity, usually written 1 or 1R. Ring homomor- phisms are assumed to map 1 to 1. A subring is assumed to have the same identity element. Usually R will denote an arbitrary ring (in this sense). Polynomial rings I will always use t, t1,...,tn to denote independent indeterminates. Thus R[t] is the ring of polynomials in one variable over R, R[t1,...,tn] is the ring of polynomials in n variables over R, etc. The most important property of these rings is the ‘universal property’, which will frequently be used without special explanation: • Given any ring homomorphism f : R → S and elements s1,...,sn ∈ S, ∗ there exists a unique ring homomorphism f : R[t1,...,tn] → S such that ∗ ∗ f (r)= f(r) ∀r ∈ R and f (ti)= si for i = 1,...,n. Modules An R-module is an abelian group M together with an action of R on M. This means: for each r ∈ R, a −→ ar (a ∈ M) is an endomorphism of the abelian group M (i.e. a homomorphism from M to itself), and moreover this assignment gives a ring homomorphism from R into EndZ(M), the ring of all additive endomorphisms of M. In practical terms this means: for all a, b ∈ M and all r, s ∈ R we have (a + b)r = ar + br a1= a a(r + s)= ar + as a(rs)=(ar)s. 1 (Here M is a right R-module; similarly one has left R-modules, but over a commutative ring these are really the same thing.) A submodule of M is an additive subgroup N such that a ∈ N, r ∈ R =⇒ ar ∈ N. -
On Orders in Separable Algebras
ON ORDERS IN SEPARABLE ALGEBRAS D. G. HIGMAN Introduction. The present note began from the observation that the arguments produced by J-M. Maranda in developing his very interesting theory of representations of groups by automorphisms of modules over Dedekind rings (4, 5) were applicable without essential change to arbitrary orders, instead of just group rings, provided that a suitable generalization of Theorem 1 of (4) could be supplied. We prove that when the ring of integers is a Dedekind ring a certain integral ideal /(©) vanishes if and only if © is an order in a separable algebra, thus extending Maranda's results to these orders and indicating that an essential change can be expected in going beyond this case. The author is indebted to Professor Maranda for the opportunity of studying (5) before its publication. Notations. The following notations will be fixed throughout. Q = Dedekind ring. K = quotient field of Q. A = (finite dimensional linear associative) algebra over K with identity element e. © = g-order in A. 1. The ideal /(©). By a two-sided G-module we shall understand a module T having © both as a ring of left and right operators such that (r«)i? = r(«^)i eu = ue = u (f, rj 6 G, u 6 T), which is finitely generated over g. For such a two-sided ©-module T we shall denote by Z(T) the g-module of all g-homomorphisms <j> of © into T such that (i) *(fr) = ww + *(r)* (r, ^®), and by B(T) the submodule of 0 € Z(T) for which there exist elements u Ç T such that (2) 0(co) = œu — uo) (w Ç ®). -
Extensions of a Ring by a Ring with a Bimodule Structure
EXTENSIONS OF A RING BY A RING WITH A BIMODULE STRUCTURE CARL W. KOHLS Abstract. A type of ring extension is considered that was intro- duced by J. Szendrei and generalizes many familiar examples, including the complex extension of the real field. We give a method for constructing a large class of examples of this type of extension, and show that for some rings all possible examples are obtained by this method. An abstract characterization of the extension is also given, among rings defined on the set product of two given rings. This paper is a sequel to [2 ], in which a class of examples was given of a type of ring extension defined in terms of two functions. Here we exhibit some other functions that may be used to construct such extensions, and show that in certain cases (in particular, when the first ring is an integral domain and the second is a commutative ring with identity), the functions must have a prescribed form. We also characterize this type of ring extension, which Szendrei defined di- rectly by the ring operations, in terms of the manner in which the two given rings are embedded in the extension. The reader is referred to [2] for background. Only the basic defini- tion is repeated here. Let A and B be rings. We define the ring A*B to be the direct sum of A and B as additive groups, with multiplication given by (1) (a,b)(c,d) = (ac+ {b,d},aad + b<rc+ bd), where a is a homomorphism from A onto a ring of permutable bimul- tiplications of B, and { •, • } is a biadditive function from BXB into A satisfying the equations (2) bo-{c,d) = o-[b.c)d, (3) {b,cd\ = {bc,d\, (4) {b,aac\ = {baa,c\, (5) \o-ab,c} = a{b,c}, and {&,c<ra} = \b,c\a, for all a£^4 and all b, c, dEB.