The concept of a skew root of a skew polynomial is used to introduce some notions of algebraic closedness for sigma-fields. By a sigma-field, we mean a field equipped with an endomorphism. It is shown that every sigma-field can be embedded in algebraically closed sigma-fields of different types. Moreover, a twisted version of the method of partial fraction decomposition is introduced and studied over algebraically closed sigma-fields.
For a commutative ring S and self-orthogonal subcategory C of Mod(S), we consider matrix factorizations whose modules belong to C. Let f in S be a regular element. If f is M-regular for every M in C, we show there is a natural embedding of the homotopy category of C-factorizations of f into a corresponding homotopy category of totally acyclic complexes. Moreover, we prove this is an equivalence if C is the category of projective or flat-cotorsion S-modules. Dually, using divisibility in place of regularity, we observe there is a parallel equivalence when C is the category of injective S-modules.
In this paper, we present a dual version of T. Brzezinski's results about Rota-Baxter systems which appeared in [Rota-Baxter systems, dendriform algebras and covariant bialgebras, J. Algebra 460 (2016) 1-25]. Then as a generalization to bialgebras, we introduce the notion of Rota-Baxter bisystem and construct various examples of Rota-Baxter bialgebras and bisystems in dimensions 2, 3 and 4. On the other hand, we introduce a new type of bialgebras (named mixed bialgebras) which consist of an associative algebra and a coassociative coalgebra satisfying the compatible condition determined by two coderivations. We investigate coquasitriangular mixed bialgebras and the particular case of coquasitriangular infinitesimal bialgebras, where we give the double construction. Also, we show in some cases that Rota-Baxter cosystems can be obtained from a coquasitriangular mixed bialgebras.
The aim of this paper is to develop the theory of Hom-coalgebras and related structures. After reviewing some key constructions and examples of quasi-deformations of Lie algebras involving twisted derivations and giving rise to the class of quasi-Lie algebras incorporating HomLie algebras, we describe the notion and some properties of Hom-algebras and provide examples. We introduce Hom-coalgebra structures, leading to the notions of Hom-bialgebra and HomHopf algebras, and prove some fundamental properties and give examples. Finally, we define the concept of HomLie admissible Hom-coalgebra and provide their classification based on subgroups of the symmetric group.
The aim of this article is to describe necessary and sufficient conditions for simplicity of Ore extension rings, with an emphasis on differential polynomial rings. We show that a differential polynomial ring, R[x;id,δ], is simple if and only if its center is a field and R is δ-simple. When R is commutative we note that the centralizer of R in R[x;σ,δ] is a maximal commutative subring containing R and, in the case when σ=id, we show that it intersects every non-zero ideal of R[x;id,δ] non-trivially. Using this we show that if R is δ-simple and maximal commutative in R[x;id,δ], then R[x;id,δ] is simple. We also show that under some conditions on R the converse holds.