Separability in algebra and category theory
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Robert Wisbauer
Abstract
Separable field extensions are essentially known since the 19th century and their formal definition was given by Ernst Steinitz in 1910. In this survey we first recall this notion and equivalent characterisations. Then we outline how these were extended to more general structures, leading to separable algebras (over rings), Frobenius algebras, (non associative) Azumaya algebras, coalgebras, Hopf algebras, and eventually to separable functors. The purpose of the talk is to demonstrate that the development of new notions and definitions can lead to simpler formulations and to a deeper understanding of the original concepts. The formalism also applies to algebras and coalgebras over semirings and S-acts (transition systems)
Abstract
Separable field extensions are essentially known since the 19th century and their formal definition was given by Ernst Steinitz in 1910. In this survey we first recall this notion and equivalent characterisations. Then we outline how these were extended to more general structures, leading to separable algebras (over rings), Frobenius algebras, (non associative) Azumaya algebras, coalgebras, Hopf algebras, and eventually to separable functors. The purpose of the talk is to demonstrate that the development of new notions and definitions can lead to simpler formulations and to a deeper understanding of the original concepts. The formalism also applies to algebras and coalgebras over semirings and S-acts (transition systems)
Chapters in this book
- Frontmatter I
- Preface V
- Contents VII
- On structure of ∗-prime rings with generalized derivation 1
- A characterization of additive mappings in rings with involution 11
- Skew constacyclic codes over Fq + vFq + v2Fq 25
- Generalized total graphs of commutative rings: a survey 37
- Differential conditions for which near-rings are commutative rings 55
- Generalized skew derivations satisfying the second posner’s theorem on lie ideals 65
- Generalized skew-derivations on lie ideals in prime rings 81
- On generalized derivations and commutativity of prime rings with involution 91
- On (n, d)-krull property in amalgamated algebra 101
- Pure ideals in ordered Γ-semigroups 111
- Projective ideals of differential polynomial rings over hnp rings 121
- Additive central m-power skew-commuting maps on semiprime rings 135
- A note on cess-lattices 151
- Properties inherited by direct sums of copies of a module 163
- Modules witnessing that a leavitt path algebra is directly infinite 181
- Inductive groupoids and normal categories of regular semigroups 193
- Actions of generalized derivations in rings and banach algebras 201
- Proper categories and their duals 215
- On nakayama conjecture and related conjectures-review 227
- On construction of global actions for partial actions 243
- On 2-absorbing and weakly 2-absorbing ideals in product lattices 253
- Separability in algebra and category theory 265
- Annihilators of power values of generalized skew derivations on lie ideals 307
- Generalized derivations on prime rings with involution 317
Chapters in this book
- Frontmatter I
- Preface V
- Contents VII
- On structure of ∗-prime rings with generalized derivation 1
- A characterization of additive mappings in rings with involution 11
- Skew constacyclic codes over Fq + vFq + v2Fq 25
- Generalized total graphs of commutative rings: a survey 37
- Differential conditions for which near-rings are commutative rings 55
- Generalized skew derivations satisfying the second posner’s theorem on lie ideals 65
- Generalized skew-derivations on lie ideals in prime rings 81
- On generalized derivations and commutativity of prime rings with involution 91
- On (n, d)-krull property in amalgamated algebra 101
- Pure ideals in ordered Γ-semigroups 111
- Projective ideals of differential polynomial rings over hnp rings 121
- Additive central m-power skew-commuting maps on semiprime rings 135
- A note on cess-lattices 151
- Properties inherited by direct sums of copies of a module 163
- Modules witnessing that a leavitt path algebra is directly infinite 181
- Inductive groupoids and normal categories of regular semigroups 193
- Actions of generalized derivations in rings and banach algebras 201
- Proper categories and their duals 215
- On nakayama conjecture and related conjectures-review 227
- On construction of global actions for partial actions 243
- On 2-absorbing and weakly 2-absorbing ideals in product lattices 253
- Separability in algebra and category theory 265
- Annihilators of power values of generalized skew derivations on lie ideals 307
- Generalized derivations on prime rings with involution 317
