Uniquely separable extensions

Let R be a commutative ring. We show that a finitely generated R-module M is reflexive (equivalently, linearly compact) if and only if R/annR(M) is reflexive (equivalently, linearly compact). If R is a quasi-noetherian ring, we prove that an R-module M is reflexive if and only if R/annR(M) is linearly compact and M has a finitely generated submodule N such that M/N is finitely cogenerated. These properly generalize results of Belshoff, Enochs and Garcia Rozas. A ring extension S of R is considered and we show that under some minor condition an S-module is reflexive if and only if it is reflexive as an R-module. Throughout this paper, rings are commutative with identity and modules are unitary. The terminology and notations not defined here can be found in Anderson and Fuller [1]. Let R be a ring and be the minimal injective cogenerator of the category of R-modules, where is an irredundant set of representatives of the simple R-modules. If M is an R-module we let . Following Belshoff, Enochs and Garcia Rozas [2], we call M a (Matlis) reflexive R-module if the evaluation map is an isomorphism. Note that eM is always injective, since U is a cogenerator of R-modules. We recall that an R-module M is linearly compact if every finitely solvable family (where each ) is solvable. For the convenience of the reader we record some basic properties of linearly compact modules and reflexive modules as follows (see [[6], Sec. 3] and [2]): (1) Every artinian module is linearly compact; (2) an infinite direct sum of non-zero modules is not linearly compact (reflexive); (3) If N is a submodule of a module M, then M is linearly compact (reflexive) if and only if both N and M/N are linearly compact (reflexive); consequently, a finite direct sum of modules is linearly compact (reflexive) if and only if each summand is linearly compact (reflexive); and (4) If M is an R-module and IM=0 for an ideal I of R, then M is linearly compact (reflexive) as an R-module if and only if it is linearly compact (reflexive) as an (R/I)-module. This paper consists of two sections. In Sec. 1, we show that the following are equivalent for a finitely generated R-module M: (1) M is reflexive; (2) M is linearly compact; (3) is linearly compact; (4) is reflexive. If R is a quasi-noetherian ring, i.e., the ideal annR(M) is finitely generated for each finitely cogenerated reflexive R-module M, we show that an R-module M is reflexive if and only if R/annR(M) is linearly compact and M has a finitely generated submodule N such that M/N is finitely cogenerated. In Sec. 2, we consider a ring extension S of R. We prove that is the minimal injective cogenerator of S-modules and an S-module M is reflexive as an S-module if and only if M is reflexive as an R-module, provided that one of the following three conditions is satisfied: (1) S is finitely generated as an R-module, (2) S is linearly compact as an R-module, or (3) S=RQ is a split extension with Q nilpotent.

Finite extensions of complex commutative Banach algebras are naturally related to corresponding finite covering maps between the carrier spaces for the algebras. In the case of function rings, the finite extensions are induced by the corresponding finite covering maps, and the topological properties of the coverings are strongly reflected in the algebraic properties of the extensions and conversely. Of particular interest to us is the class of finite covering maps for which the induced extensions of function rings admit primitive generators. This is exactly the class of polynomial covering maps and the extensions are algebraic extensions defined by the underlying Weierstrass polynomials.The purpose of this paper is to develop a suitable Galois theory for finite extensions of function rings induced by finite covering maps and to apply it in the case of Weierstrass polynomials and polynomial covering maps.

An algebra extension A ∣ B is right depth two if its tensor-square \(A \otimes _{B} A\) is in the Dress category \({\mathbf{Add}}_{A} A_{B}\). We consider necessary conditions for right, similarly left, D2 extensions in terms of partial A-invariance of two-sided ideals in A contracted to the centralizer. Finite dimensional algebras extending central simple algebras are shown to be depth two. Following P. Xu, left and right bialgebroids over a base algebra R may be defined in terms of anchor maps, or representations on R. The anchor maps for the bialgebroids \(S = {\text{End}}_{B} A_{B}\) and \(T = {\text{End}}_{A} A \otimes _{B} A_{A}\) over the centralizer R = C A (B) are the modules S R and R T studied in Kadison (J. Alg. & Appl., 2005, preprint), Kadison (Contemp. Math., 391: 149–156, 2005), and Kadison and Külshammer (Commun. Algebra, 34: 3103–3122, 2006), which provide information about the bialgebroids and the extension (Kadison, Bull. Belg. Math. Soc. Simon Stevin, 12: 275–293, 2005). The anchor maps for the Hopf algebroids in Khalkhali and Rangipour (Lett. Math. Phys., 70: 259–272, 2004) and Kadison (2005, preprint) reverse the order of right multiplication and action by a Hopf algebra element, and lift to the isomorphism in Van Oystaeyen and Panaite (Appl. Categ. Struct., 2006, in press). We sketch a theory of stable A-modules and their endomorphism rings and generalize the smash product decomposition in Kadison (Proc. Am. Math. Soc., 131: 2993–3002, 2003 Prop. 1.1) to any A-module. We observe that Schneider’s coGalois theory in Schneider (Isr. J. Math., 72: 167–195, 1990) provides examples of codepth two, such as the quotient epimorphism of a finite dimensional normal Hopf subalgebra. A homomorphism of finite dimensional coalgebras is codepth two if and only if its dual homomorphism of algebras is depth two.

The Baer and the quasi-Baer properties of rings are extended to a module theoretic setting in this chapter. Using the endomorphism ring of a module, the notions of Baer, quasi-Baer, and Rickart modules are introduced and studied. Similar to the fact that every Baer ring is nonsingular, we shall see that every Baer module satisfies a weaker notion of nonsingularity of modules (\(\mathcal{K}\)-nonsingularity) which depends on the endomorphism ring of the module. Strong connections between a Baer module and an extending module will be observed via this weak nonsingularity and its dual notion. It is shown that an extending module which is \(\mathcal{K}\)-nonsingular is precisely a \(\mathcal{K}\)-cononsingular Baer module. This provides a module theoretic analogue of the Chatters-Khuri theorem for rings. Direct summands of Baer and quasi-Baer modules respectively inherit these properties. This provides a rich source of examples of Baer and quasi-Baer modules, since one can readily see that for any (quasi-)Baer ring R and an idempotent e in R, the right R-module eR R is always a (quasi-)Baer module. It will be seen that every projective module over a quasi-Baer ring is a quasi-Baer module. Connections of a (quasi-)Baer module and its endomorphism ring are discussed. Characterizations of classes of rings via the Baer property of certain classes of free modules over them are presented. An application also yields a type theory for \(\mathcal{K}\)-nonsingular extending (continuous) modules which, in particular, improve the type theory for nonsingular injective modules provided by Goodearl and Boyle.

It seems that Morita invariance judges of the importance of classes of ring extensions concerned. Miyashita introduced the notion of Morita equivalence in ring extensions, and he showed that the classes of G-Galois extensions and Frobenius extensions are Morita invariant. After that, Ikehata showed that the classes of separable extensions, Hirata separable extensions, symmetric extensions, and QF-extensions are Morita invariant. In this article, we shall prove that the classes of several extensions are Morita invariant. Further, we will give an example of the class of ring extensions which is not Morita invariant.

While every finite lattice-based algebra is dualisable, the same is not true of semilattice-based algebras. We show that a finite semilattice-based algebra is dualisable if all its operations are compatible with the semilattice operation. We also give examples of infinite semilattice-based algebras that are dualisable. In contrast, we present a general condition that guarantees the inherent non-dualisability of a finite semilattice-based algebra. We combine our results to characterise dualisability amongst the finite algebras in the classes of flat extensions of partial algebras and closure semilattices. Throughout, we emphasise the connection between the dualisability of an algebra and the residual character of the variety it generates.

Torsion for abelian varieties of type III

In this paper, the complete algebraic structure of the finite semisimple group algebra of a normally monomial group is described. The main result is illustrated by computing the explicit Wedderburn decomposition of finite semisimple group algebras of various normally monomial groups. The automorphism groups of these group algebras are also determined.

We show that every central simple algebra $A$ over a field $k$ is Brauer equivalent to a quotient of a finite dimensional Hopf algebra over the same field. This shows that the natural generalization of the Schur group for Hopf algebras (which we call the Hopf Schur group) is in fact the entire Brauer group of $k$. If the characteristic of the field is zero, or if the algebra has a Galois splitting field with certain properties, we can take this Hopf algebra to be semisimple. We also show that if $F$ is any finite separable extension of $k$, then $F$ is a quotient of a finite dimensional commutative semisimple and cosemisimple Hopf algebra over $k$.

In this chapter we shall analyze arbitrary extension fields of a field Φ. A study of finite dimensional extension fields and a partial study of algebraic extensions has been made in Chapter I. In this chapter our primary concern will be with infinite dimensional extensions and we shall begin again with the algebraic ones. We define algebraically closed fields and prove the existence of an algebraic closure of any field. We shall extend the classical Galois theory to apply to infinite dimensional normal and separable extensions. After this we shall consider arbitrary extension fields and we shall show that these can be built up in two stages: first a purely transcendental one and then on top of this an algebraic extension. The invariant of this mode of generating a field is the transcendency degree which is the cardinal number of a transcendency basis. We shall obtain conditions for the existence of a transcendency basis such that the extension is separable algebraic over the purely transcendental extension determined by the basis. We shall also give a definition of separability of an extension field that generalizes the notion of algebraic separability. The notion of a derivation plays an important role in these considerations.KeywordsPrime IdealStructure TheoryGalois GroupHigh DerivationAlgebraic ClosureThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Symmetry reduction for the Kadomtsev-Petviashvili equation using its infinite dimensional invariance group

The representation theory of semisimple algebraic groups over the complex numbers (equivalently, semisimple complex Lie algebras or Lie groups, or real compact Lie groups) and the questions of whether a given complex representation is symplectic or orthogonal have been solved since at least the 1950s. Similar results for Weyl modules of split reductive groups over fields of characteristic different from z hold by using similar proofs. This paper considers analogues of these results for simple, induced, and tilting modules of split reductive groups over fields of prime characteristic as well as a complete answer for Weyl modules over fields of characteristic 2.

M. E. Sweedler first constructed a universal Hopf algebra of an algebra. It is known that the dual notions to the existing ones play a dominant role in Hopf algebra theory. Yu. I. Manin and D. Tambara introduced the dual notion of Sweedler's construction in separate works. In this paper, we construct a universal algebra for a finite-dimensional Lie-Yamaguti algebra. We demonstrate that this universal algebra possesses a bialgebra structure, leading to a universal coacting Hopf algebra for a finite-dimensional Lie-Yamaguti algebra. Additionally, we develop a representation-theoretic version of our results. As an application, we characterize the automorphism group and classify all abelian group gradings of a finite-dimensional Lie-Yamaguti algebra.

Centralizers in semisimple algebras, and descent spectrum in Banach algebras

Let a ring A be an H-separable extension of a subring B of A, that is, A ⊗ B A is an A-A-direct summand of a finite direct sum of copies of A. If furthermore, (a) B is a left B-direct summand of A (or (b) A is left B-finitely generated projective), and if B is a right (resp. left) artinian QF-3 ring, then A is also a right (resp. left) artinian QF-3 ring. Consequently, if A is an H-separable extension of a serial ring B with one of the conditions (a), (b), then A is also a serial ring. In particular H-separable extension of a uni-serial ring is always uni-serial.