Robert Wisbauer

Foundations of module and ring theory : a handbook for study and research

This volume provides a comprehensive introduction to module theory and the related part of ring theory, including original results as well as the most recent work. It is a useful and stimulating study for those new to the subject as well as for researchers and serves as a reference volume. Starting form a basic understanding of linear algebra, the theory is presented and accompanied by complete proofs. For a module M, the smallest Grothendieck category containing it is denoted by o[M] and module theory is developed in this category. Developing the techniques in o[M] is no more complicated than in full module categories and the higher generality yields significant advantages: for example, module theory may be developed for rings without units and also for non-associative rings. Numerous exercises are included in this volume to give further insight into the topics covered and to draw attention to related results in the literature.

Monads and comonads on module categories

Let A be a ring and MA the category of right A-modules. It is well known in module theory that any A-bimodule B is an A-ring if and only if the functor −⊗AB:MA→MA is a monad (or triple). Similarly, an A-bimodule C is an A-coring provided the functor −⊗AC:MA→MA is a comonad (or cotriple). The related categories of modules (or algebras) of −⊗AB and comodules (or coalgebras) of −⊗AC are well studied in the literature. On the other hand, the right adjoint endofunctors HomA(B,−) and HomA(C,−) are a comonad and a monad, respectively, but the corresponding (co)module categories did not find much attention so far. The category of HomA(B,−)-comodules is isomorphic to the category of B-modules, while the category of HomA(C,−)-modules (called C-contramodules by Eilenberg and Moore) need not be equivalent to the category of C-comodules. The purpose of this paper is to investigate these categories and their relationships based on some observations of the categorical background. This leads to a deeper understanding and characterisations of algebraic structures such as corings, bialgebras and Hopf algebras. For example, it turns out that the categories of C-comodules and HomA(C,−)-modules are equivalent provided C is a coseparable coring. Furthermore, we describe equivalences between categories of HomA(C,−)-modules and comodules over a coring D in terms of new Galois properties of bicomodules. Finally, we characterise Hopf algebras H over a commutative ring R by properties of the functor HomR(H,−) and the category of mixed HomR(H,−)-bimodules. This generalises in particular the fact that a finite dimensional vector space H is a Hopf algebra if and only if the dual space H∗ is a Hopf algebra.

Algebras Versus Coalgebras

Algebras and coalgebras are fundamental notions for large parts of mathematics. The basic constructions from universal algebra are now expressed in the language of categories and thus are accessible to classical algebraists and topologists as well as to logicians and computer scientists. Some of them have developed specialised parts of the theory and often reinvented constructions already known in a neighbouring area. One purpose of this survey is to show the connection between results from different fields and to trace a number of them back to some fundamental papers in category theory from the early 1970s. Another intention is to look at the interplay between algebraic and coalgebraic notions. Hopf algebras are one of the most interesting objects in this setting. While knowledge of algebras and coalgebras are folklore in general category theory, the notion of Hopf algebras is usually only considered for monoidal categories. In the course of the text we do suggest how to overcome this defect by defining a Hopf monad on an arbitrary category as a monad and comonad satisfying some compatibility conditions and inducing an equivalence between the base category and the category of the associated bimodules. For a set G, the endofunctor G× – on the category of sets shares these properties if and only if G admits a group structure. Finally, we report about the possibility of subsuming algebras and coalgebras in the notion of (F,G)-dimodules associated to two functors

On Galois Comodules

F,G:\mathbb{A}\to \mathbb{B}

Dual coalgebras of algebras over commutative rings

between different categories. This observation, due to Tatsuya Hagino, was an outcome from the theory of categorical data types and may also be of use in classical algebra.

Unitary strongly prime rings and related radicals

Generalizing the notion of Galois corings, Galois comodules were introduced as comodules P over an A-coring 𝒞 for which P A is finitely generated and projective and the evaluation map μ𝒞:Hom 𝒞 (P, 𝒞) ⊗ S P → 𝒞 is an isomorphism (of corings) where S = End 𝒞 (P). It has been observed that for such comodules the functors − ⊗ A 𝒞 and Hom A (P, −) ⊗ S P from the category of right A-modules to the category of right 𝒞-comodules are isomorphic. In this note we use this isomorphism related to a comodule P to define Galois comodules without requiring P A to be finitely generated and projective. This generalises the old notion with this name but we show that essential properties and relationships are maintained. Galois comodules are close to being generators and have common properties with tilting (co)modules. Some of our results also apply to generalised Hopf Galois (coalgebra Galois) extensions.

On the category of comodules over corings

Abstract In the study of algebraic groups the representative functions related to monoid algebras over fields provide an important tool which also yields the finite dual coalgebra of any algebra over a field. The purpose of this note is to transfer this basic construction to monoid algebras over commutative rings R. As an application we obtain a bialgebra (Hopf algebra) structure on the finite dual of the polynomial ring R[x] over a noetherian ring R. Moreover, we give a sufficient condition for the finite dual of any R-algebra A to become a coalgebra. In particular, this condition is satisfied provided R is noetherian and hereditary.

ON P-INJECTIVE RINGS

A unitary strongly prime ring is defined as a prime ring whose central closure is simple with identity element. The class of unitary strongly prime rings is a special class of rings and the corresponding radical is called the unitary strongly prime radical. In this paper we prove some results on unitary strongly prime rings. The results are applied to study the unitary strongly prime radical of a polynomial ring and also R-disjoint maximal ideals of polynomial rings over R in a finite number of indeterminates. From this we get relations between the Brown–McCoy radical and the unitary strongly prime radical of polynomial rings. In particular, the Brown–McCoy radical of R[X] is equal to the unitary strongly prime radical of R[X] and also equal to S(R)[X], where S(R) denotes the unitary strongly prime radical of R, when X is an infinite set of either commuting or non-commuting indeterminates. For a PI ring R this holds for any set X.

Properly semiprime self-pp-modules

It is well known that the category MC of right comodules over an A-coring C, A an associative ring, is a subcategory of the category of left modules ∗CM over the dual ring ∗C. The main purpose of this note is to show that MC is a full subcatgeory in ∗CM if and only if C is locally projective as a left A-module.

On Coprime Modules and Comodules

1. Definitions and preliminary results. Throughout this paper R will be an associative ring with unity and all /?-modules are unitary. The right (resp. left) annihilator in R of a subset X of a module is denoted by r(X) (resp. \(X)). The Jacobson radical of R is denoted by J(R), the singular ideals are denoted by Z(RR) and Z{RR) and the socles by Soc(RR) and Soc(RR). For a module M, E(M) and PE(M) denote the injective and pure-injective envelopes of M, respectively. For a submodule A^M, the notation A c® M will mean that A is a direct summand of M. A module MR is called p-injective if for every a e R, every 7?-linear map from aR to A/ can be extended to an ^-linear map from R to M. R is called right p-injective if RR is p-injective. Recall that a module MR is called uniserial if its submodules are linearly ordered by inclusion and serial if it is a direct sum of uniserial submodules. A ring R is right uniserial (serial) if RR is uniserial (serial). We record some well-known results on serial and p-injective rings.