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9 8 7 6 5 4 3 2 1 www.birkhauser.ch Contents

Preface ...... ix

T. Albu Applications of Cogalois Theory to Elementary Arithmetic . . . . . 1

A. Alvarado Garc´ıa,H.A. Rinc´onMej´ıa and J. R´ıosMontes On Big Lattices of Classes of R-modules Defined byClosureProperties ...... 19

H.E. Bell and Y. Li ReversibleandDuoGroupRings ...... 37

G.F. Birkenmeier, J.K. Park and S.T. Rizvi PrincipallyQuasi-BaerRingHulls ...... 47

G.L. Booth Strongly Prime Ideals of Near-rings of Continuous Functions ...... 63

W.D. Burgess, A. Lashgari and A. Mojiri ElementsofMinimalPrimeIdealsinGeneralRings ...... 69

V. Camillo and P.P. Nielsen Ona TheoremofCampsandDicks ...... 83

M.M. Choban and M.I. Ursul Applications of the Stone Duality in theTheoryofPrecompactBooleanRings ...... 85

J. Dauns OverRingsandFunctors ...... 113

H.Q. Dinh On Some Classes of Repeated-root Constacyclic Codes ofLengtha Powerof2 overGaloisRings ...... 131

A. Facchini and N. Girardi CouniformlyPresentedModulesandDualities ...... 149 vi Contents

K.R. Goodearl SemiclassicalLimitsofQuantizedCoordinateRings ...... 165

D. Khurana, G. Marks and A.K. Srivastava OnUnit-CentralRings ...... 205

T.Y. Lam and R.G. Swan Symplectic Modules and von Neumann Regular Matrices overCommutativeRings ...... 213

G. Marks and M. Schmidmeier Extensions of Simple Modules and the Converse ofSchur’sLemma ...... 229

S.H. Mohamed ReportonExchangeRings ...... 239

D.S. Passman FiltrationsinSemisimpleLieAlgebras,III ...... 257

D.P. Patil OntheBlowing-upRings,ArfRingsandTypeSequences ...... 269

Z. Izhakian and L. Rowen A GuidetoSupertropicalAlgebra ...... 283

P.F. Smith Projective Modules, Idempotent Ideals andIntersectionTheorems ...... 303

L.V. Thuyet and T.C. Quynh On Ef-extending Modules and Rings withChainConditions ...... 327

Y. Zhou OnCleanGroupRings ...... 335 S.K. Jain Preface

The International Conference on Algebra and its Applications held in Athens, Ohio, June 18–21, 2008 and sponsored by the Ohio University Center for Ring Theory and its Applications (CRA) had as its central purpose to honor Surender K. Jain, the Center’s retiring first director, on the dual occasion of his 70th birthday and of his retirement from Ohio University. With this volume we celebrate the contributions to Algebra of our distinguished colleague. One of Surender’s main attributes has been the way in which he radiates enthusiasm about mathematical research; his eagerness to pursue mathematical problems is contagious; we hope that reading this excellent collection of scholarly writings will have a similar effect on our readers and that you will be inspired to continue the pursuit of Ring Theory as well as Algebra and its Applications. As with previous installments of CRA conferences, the underlying principle behind the meeting was to bring together specialists on the various areas of Al- gebra in order to promote communication and cross pollination between them. In particular, a common philosophy of our conferences through the years has been to bring algebraists who focus on the theoretical aspects of our field with those others who embrace applications of Algebra in diverse areas. Clearly, as a reflection of the interests of the organizers, the applications we emphasized were largely within the realm of Coding Theory. The philosophy behind the organization of the conference has undoubtedly impacted this Proceedings volume. For the most part, the contributors delivered related talks at the conference itself. However, there are also a couple of contributions in this volume from authors who could not be present at the conference but wanted to participate and honor Dr. Jain on this occasion. All papers were subject to a strict process of refereeing and, in fact, not all submissions were accepted for publication. We would like to take this opportunity to thank all the anonymous refer- ees who delivered their verdicts about the submitted papers within a very tight schedule; they also provided valuable feedback on many of the papers that appear here in final form. Likewise, we wish to express our deep appreciation to Sylvia Lotrovsky and Thomas Hempfling of Birkh¨auser for their diligent efforts to bring this volume to completion. Advances in Ring Theory Trends in Mathematics, 1–17 c 2010 Birkh¨auser Verlag Basel/Switzerland

Applications of Cogalois Theory to Elementary Field Arithmetic

Toma Albu

Dedicated to S.K. Jain on his 70th birthday

Abstract. The aim of this expository paper is to present those basic concepts and facts of Cogalois theory which will be used for obtaining in a natural and easy way some interesting results in elementary field arithmetic. Mathematics Subject Classification (2000). Primary 12-06, 12E30, 11-06, 11A99; Secondary 12F05, 12F10, 12F99, 12Y05. Keywords. Cogalois theory, elementary field arithmetic, field extension, , radical extension, Kneser extension, Cogalois extension, G-Cogalois extension.

1. Introduction A standard, very concrete, and not so hard exercise in any undergraduate abstract algebra course anyone of us has encountered is the following one: √ √ Q ⊆ Q 3 Consider the field extension √ √( 2, 5). (a) Calculate the degree [ Q( 2, 3 5) : Q ] of this extension. (b) Find a primitive element of this extension.

Surely, it is natural to ask√ what about√ the same questions when we replace the very particular radicals 2and 3 5 by arbitrary finitely many radicals of positive integers. More precisely, we have the following Problem. Consider the field extension √ √ n1 nr Q ⊆ Q ( a1 ,..., ar ),

The author gratefully acknowledges partial financial support from the grant ID-PCE 1190/2008 awarded by the Consiliul Nat¸ional al Cercet˘arii S¸tiint¸ifice ˆın ˆInv˘at¸˘amˆantul Superior (CNCSIS), Romˆania. 2T.Albu √ ni where r, n1,...,nr,a1,...,ar are positive integers, and where ai is the positive real nith root of ai for each i, 1  i  r. √ √ n1 nr (a) Calculate the degree [ Q ( a1 ,..., ar ):Q ] of this extension. (b) Find a primitive element of this extension.

More than twenty years ago we first thought about this challenging problem. A first attempt to solve it, even in a more general case, was the introduction and investigation of the so-called Kummer extensions with few roots of unity,seeAlbu [1]. After that, we discovered, little by little, the fundamental papers of Kneser [25] and Greither and Harrison [20] and got more and more involved in their topic, which lead to what is nowadays called Cogalois theory.Thereareatleasttwo reasons for presenting this material to ring and module theorists:

• firstly, to make a propaganda of this pretty nice and equally new theory in field theory by providing a gentle and as short as possible introduction to a general audience and readership of its basic notions and results, and • secondly, we want to show how this theory has nice applications in solv- ing some interesting and nontrivial problems of elementary field arithmetic, including that mentioned above concerning the computation of the degree Q ⊆ and√ finding a√ (canonical) primitive element of field extensions like n1 nr Q ( a1 ,..., ar ).

2. Notation and terminology By N we denote the set {0, 1, 2,...} of all natural numbers, by N∗ the set N\{0} of all strictly positive natural numbers, and by Q (resp. R, C) the field of all rational (resp. real, complex) numbers. For any ∅ = A ⊆ C (resp. ∅ = X ⊆ R ) ∗ ∗ ∗ we denote A = A \{0} (resp. X+ = { x ∈ X | x  0 }). If a ∈ R+ and n ∈ N , n then√ the unique positive real root of the equation x − a = 0 will be denoted by n a. For any set M, |M| will denote the cardinal number of M. A field extension is a pair (F, E) of fields, where F is a subfield of E (or E is an overfield of F ),andinthiscaseweshallwriteE/F. Very often, instead of “field extension” we shall use the shorter term “extension”. If E is an overfield of afieldF , we will also say that E is an extension of F .Byanintermediate field of an extension E/F we mean any subfield K of E with F ⊆ K, and the set of all intermediate fields of E/F is a complete lattice that will be denoted by I(E/F). Throughout this paper F always denotes a field, Char(F ) its characteristic, e(F ) its characteristic exponent (that is, e(F )=1ifF has characteristic 0, and e(F )=p if F has characteristic p>0), and Ω a fixed algebraically closed field containing F as a subfield. Any considered overfield of F is supposed to be a subfield of Ω. Applications of Cogalois Theory 3

For an arbitrary nonempty subset S of Ω and a number n ∈ N∗ we denote throughout this paper: S∗ = S \{0}, Sn = { xn | x ∈ S }, n μn(S)={ x ∈ S | x =1}.

By a primitive nth root of unity we mean any generator of the μn(Ω); ζn will always denote such an element. For an arbitrary group G, the notation H  G means that H is a subgroup of G. The lattice of all subgroups of G will be denoted by L(G). For any subset M of G, M will denote the subgroup of G generated by M. For a field extension E/F we shall denote by [E : F ]thedegree,andby Gal (E/F)theGalois group of E/F. For any subgroup Δ of Gal (E/F), Fix (Δ) will denote the fixed field of Δ. If E/F is an extension and A ⊆ E,thenF [A] will denote the smallest subring of E containing both A and F as subsets. We also denote by F (A) the smallest subfield of E containing both A and F as subsets, called the subfield of E obtained by adjoining to F the set A. For all other undefined terms and notation concerning basic field theory the reader is referred to Bourbaki [17], Karpilovsky [24], and/or Lang [26].

3. What is Cogalois theory? Cogalois theory, a fairly new area in field theory, investigates field extensions, finite or not, that possess a so-called Cogalois correspondence. The subject is somewhat dual to the very classical Galois theory dealing with field extensions possessing a Galois correspondence. In what follows we are intending to briefly explain the meaning of such ex- tensions. An interesting but difficult problem in field theory is to describe in a satisfactory manner the set I(E/F) of all intermediate fields of a given field ex- tension E/F, which, in general is a complicated-to-conceive, potentially infinite set of hard-to-describe-and-identify objects. This is a very particular case of a more general problem in mathematics: Describe in a satisfactory manner the collection Sub(X) of all subobjects of a given object X of a category C. For instance, if G is a group, then an important problem in group theory is to describe the set L(G) of all subgroups of G. Observe that for any field F we may consider the category EF of all field extensions of F .IfE is any object of EF , i.e., a field extension E/F, then the set I(E/F) of all subfields of E containing F , i.e., of all intermediate fields of E/F, is precisely the set Sub(E) of all subobjects of E in EF . Another important problem in field theory is to calculate the degree of a given field extension E/F. Answers to these two problems are given for particular field extensions by Galois theory invented by E. Galois (1811–1832) and by Kummer theory invented 4T.Albu by E. Kummer (1810–1873). Let us briefly recall the solutions offered by these two theories in answering the two problems presented above. The fundamental theorem of finite Galois theory (FTFGT). If E/F is a finite Galois extension with Γ, then the canonical map α : I(E/F) −→ L(Γ),α(K)=Gal(E/K), is a lattice anti-isomorphism, i.e., a bijective order-reversing map. Moreover, [E : F ]=|Γ|. We say that such an E/F is an extension with Γ-Galois correspondence. In this way, the lattice I(E/F) of all subobjects of an object E ∈EF ,which has the additional property that is a finite Galois extension of F , can be described by the lattice of all subobjects of the object Gal (E/F) in the category Gf of all finite groups. In principle, this category is more suitable than the category EF of all field extensions of F , since the set of all subgroups of a finite group is a far more benign object. Thus, many questions concerning a field are best studied by transforming them into group theoretical questions in the group of automorphisms of the field. Note that for an infinite Galois extension E/F the FTFGT fails. In this case the Galois group Gal (E/F)isinfactaprofinite group, that is, a projective limit of finite groups, or equivalently, a Hausdorff, compact, totally disconnected topological group; its topology is the so called Krull topology. The description of I(E/F)isgivenby The fundamental theorem of infinite Galois theory (FTIGT). If E/F is an arbitrary Galois extension with Galois group Γ, then the canonical map α : I(E/F) −→ L(Γ),α(K)=Gal(E/K), is a lattice anti-isomorphism, where L(Γ) denotes the lattice of all closed subgroups of the group Γ endowed with the Krull topology. Observe that the lattice L(Γ) is nothing else than the lattice of all subobjects of Γ in the category of all profinite groups. However, the Galois group of a given Galois field extension E/F, finite or not, is in general difficult to be concretely described; so, it will be desirable to impose additional conditions on E/F such that the lattice I(E/F) be isomorphic (or anti- isomorphic) to the lattice L(Δ) of all subgroups of some other group Δ, easily computable and appearing explicitly in the data of the given Galois extension E/F. A class of such Galois extensions is that of classical Kummer extensions. We recall their definition below. Definition. A field extension E/F is said to be a classical n-Kummer extension, with n a given positive integer, if the following three conditions are satisfied: (1) gcd(n, e(F )) = 1, ∈ (2) ζn F , √ n (3) E = F ({ ai | i ∈ I }), Applications of Cogalois Theory 5 √ ∗ n where I is an arbitrary set, finite or not, ai ∈ F ,and ai is a certain root in n Ω of the polynomial X − ai, i ∈ I. Note that the extension E/F is finite if and only if the set I in the definition above can be chosen to be finite. For a classical n-Kummer extension E/F we denote by √ ∗ n ∗ Kum(E/F):=F { ai | i ∈ I /F the so-called Kummer group of E/F. The next result is a part of the so-called Kummer theory. The fundamental theorem of Kummer theory (FTKT). Let E/F be a classical n-Kummer extension with Kummer group Δ. Then there exists a canonical lattice isomorphism ∼ I(E/F) −→ L(Δ).

Observe that the Kummer group Δ of a classical n-Kummer extension E/F is intrinsically given with the extension E/F and easily manageable as well. This group is isomorphic, but not canonically, with the character group Γ of the Galois group Γ of E/F; in particular, it follows that for E/F finite, the group Δ is isomorphic with Γ, and in particular it has exactly [E : F ] elements.√ Consequently,√ n n if E/F is a finite classical n-Kummer extension, say E = F ( a1 ,..., ar ), then √ √ √ √ n n ∗ n n ∗ [ F ( a1 ,..., ar ):F ]=|F  a1 ,..., ar /F |. Note also that any classical n-Kummer extension E/F is a Galois extension with an Abelian Galois group of exponent a divisor of n (this means that σn = 1E for all σ ∈ Gal(E/F)), and conversely, any Galois extension E/F such that ∗ gcd(n, e(F )) = 1,ζn ∈ F for some n ∈ N , and such that the Galois group of E/F is an Abelian group of exponent a divisor of n, is a classical n-Kummer extension. On the other hand, there exists a fairly large class of field extensions which are not necessarily Galois, but enjoy a property similar to that in FTKT or is dual to that in FTFGT. Namely, these are the extensions E/F for which there exists a canonical lattice isomorphism (and not a lattice anti-isomorphism as in the Galois case) between I(E/F)andL(Δ), where Δ is a certain group canonically associated with the extension E/F.Wecallthemembersofthisclassextensions with Δ-Cogalois correspondence. Their prototype is the field extension √ √ n1 nr Q ( a1 ,..., ar )/Q , √ ni where r, n1,...,nr,a1,...,ar are positive integers, and where ai is the posi-   tive real nith root of ai for each i,√1 i r. √For such an extension, the associated ∗ n1 nr ∗ group Δ is the factor group Q  a1 ,..., ar / Q . Note that the finite clas- sical n-Kummer extensions have a privileged position: they are at the same time extensions with Galois and with Cogalois correspondences, and the two groups appearing in this setting are isomorphic. 6T.Albu

After 1930 there were attempts to weaken the condition ζn ∈ F in the defini- tion of a Kummer extension in order to effectively compute√ the degree√ of particular n1 nr finite radical extensions, i.e., of extensions of type F ( a1 ,..., ar )/F ,where F was mainly an algebraic number field. All these attempts finally lead to what nowadays is called Cogalois theory, also spelled co-Galois theory. The main precursors of Cogalois theory, in chronological order, are H. Hasse (1930), A. Besicovitch (1940) [16] , L.J. Mordell (1953) [27], C.L. Siegel (1972) [29], M. Kneser (1975) [25] whose paper brilliantly superseded all the previous work done in computing the degree of finite radical extensions, A. Schinzel (1975) [28],D.Gay,W.Y.V´elez (1978) [19], etc. In our opinion, Cogalois theory was born in 1986, with birthplace Journal of Pure and Applied Algebra [20], and having C. Greither and D.K. Harrison as parents. In that paper [20], the Cogalois extensions have been introduced and investigated for the first time in the literature, and other classes of finite field extensions possessing a Cogalois correspondence, including the so-called neat pre- sentations have been considered. Besides the Cogalois extensions introduced by Greither and Harrison [20] in 1986, new basic classes of finite radical field extensions the Cogalois theory deals with, namely the G-Kneser extensions, strongly G-Kneser extensions,and G-Cogalois extensions were introduced and investigated in 1995 by T. Albu and F. Nicolae [9]. Note that the frame of G-Cogalois extensions permits a simple and unified manner to study the classical Kummer extensions, the Kummer extensions with few roots of unity, the Cogalois extensions, and the neat presentations. In 2001 an infinite Cogalois theory investigating infinite radical extensions has been developed by T. Albu and M. T¸ ena, in 2003 appeared the author’s monograph “Cogalois theory” [7], and in 2005 the infinite Cogalois theory has been generalized to arbitrary profinite groups by T. Albu and S¸.A. Basarab [8], leading to a so-called abstract Cogalois theory for arbitrary profinite groups. Roughly speaking, Cogalois theory investigates√ radical extensions, finite or ni ∗ ∗ not, i.e., extensions of type E/F with E = F ({ ai | i ∈ I }), ni ∈ N ,ai ∈ F ,i∈ I, I an arbitrary set, finite or not, such that there exists a lattice isomorphism

∼ I(E/F) −→ L(Δ), where Δ is√ a group canonically associated with the given extension E/F.Mostly, ∗ ni ∗ Δ=F { ai | i ∈ I /F .

4. Basic concepts and results of Cogalois theory In this section we will briefly present some of the basic notions and facts of Co- galois theory, namely those of G-radical extension, G-Kneser extension, Cogalois extension, strongly G-Kneser extension,andG-Cogalois extension.