In this article, we shall survey some aspects of our recent (or related) constructions of derived equivalences for algebras and rings.

We define the right regular dual of an object X in a monoidal category C , and give several results regarding the weak rigid monoidal category. Based on the definition of the right regular dual, we construct a weak Hopf algebra structure of H = End(F) whenever (F, J) is a fiber functor from category C to V ec and every X ∈ C has a right regular dual. To conclude, we give a weak reconstruction theorem for a kind of weak Hopf algebra.

Let D be a finite-dimensional integral domain, Spec(D) be the set of prime ideals of D, and SpSS(D) be the set of spectral semistar operations on D. Mimouni gave a complete description for the prime ideal structure of D with |SpSS(D)| = n + dim(D) for 1≤n≤5 except for the quasi-local cases of n = 4, 5. In this paper, we show that there is an integral domain D such that |SpSS(D)| = n+dim(D) for all positive integers n with n ≠2. As corollaries, we completely characterize the quasi-local domains D with |SpSS(D)| = n+dim(D) for n = 4, 5. Furthermore, we also present the lower and upper bounds of |SpSS(D)| when Spec(D) is a finite tree.

We consider whether the tilting properties of a tilting A-module T and a tilting B-module T ^' can convey to their tensor product T ⊗T ^'. The main result is that T ⊗ T ^' turns out to be an (n+ m)-tilting A ⊗ B-module, where T is an m-tilting A-module and T ^' is an n-tilting B-module.

In this paper, the notion of a twisted partial Hopf coaction is introduced. The conditions on partial cocycles are established in order to construct partial crossed coproducts. Then the classification of partial crossed coproducts is discussed. Finally, some necessary and sufficient conditions for a class of partial crossed coproducts to be quasitriangular bialgebras are given.

We define Gorenstein injective quasi-coherent sheaves, and prove that the notion is local in case the scheme is Gorenstein. We also give a new characterization of a Gorenstein scheme in terms of the total acyclicity of every acyclic complex of injective quasi-coherent modules.

Let R be any ring. We motivate the study of a class of chain complexes of injective R-modules that we call AC-injective complexes, showing that K(AC-Inj), the chain homotopy category of all AC-injective complexes, is always a compactly generated triangulated category. In general, all DGinjective complexes are AC-injective and in fact there is a recollement linking K(AC-Inj) to the usual derived category D(R). This is based on the author’s recent work inspired by work of Krause and Stovicek. Our focus here is on giving straightforward proofs that our categories are compactly generated.

Let R be a ring with identity. We use J(R), G(R), and X(R) to denote the Jacobson radical, the group of all units, and the set of all nonzero nonunits in R, respectively. A ring is said to be Abelian if every idempotent is central. It is shown, for an Abelian ring R and an idempotent-lifting ideal N ⊆ J(R) of R, that R has a complete set of primitive idempotents if and only if R/N has a complete set of primitive idempotents. The structure of an Abelian ring R is completely determined in relation with the local property when X(R) is a union of 2, 3, 4, and 5 orbits under the left regular action on X(R) by G(R). For a semiperfect ring R which is not local, it is shown that if G(R) is a cyclic group with 2 ∈ G(R), then R is finite. We lastly consider two sorts of conditions for G(R) to be an Abelian group.

Recently, we introduced the notion of a generalized derivation from a bimodule to a bimodule. In this paper, we give a more general notion based on commutators which covers generalized derivations as a special case. Using it, we show that the separability of an algebra extension is characterized by generalized derivations.

In this paper, for rings R, we introduce complex rings ℂ(R), quaternion rings ℍ(R), and octonion rings О, which are extension rings of R; R ⊂ ℂ(R) ⊂ ℍ(R) ⊂ O(R). Our main purpose of this paper is to show that if R is a Frobenius algebra, then these extension rings are Frobenius algebras and if R is a quasi-Frobenius ring, then ℂ(R) and ℍ(R) are quasi-Frobenius rings and, when Char(R) = 2, O(R) is also a quasi-Frobenius ring.

Let R be a graded ring. We define and study strongly Gorenstein gr-projective, gr-injective, and gr-flat modules. Some connections among these modules are discussed. We also explore the relations between the graded and the ungraded strongly Gorenstein modules.

*Abstract:Let (H,α) be a monoidal Hom-Hopf algebra. In this paper, we will study the category of Hom-Yetter-Drinfeld modules. First, we show that the category of left-left Hom-Yetter-Drinfeld modules HH H? Y? D is isomorphic to the center of the category of left (H,α)-Hom-modules. Also, by the center construction, we get that the categories of left-left, left-right, right-left, and right-right Hom-Yetter-Drinfeld modules are isomorphic as braided monoidal categories. Second, we prove that the category of finitely generated projective left-left Hom-Yetter-Drinfeld modules has left and right duality.

Let R be an associative ring with identity. An R-module M is called an NCS module if C(M)∩S(M)={0}, where C(M) and S(M) denote the set of all closed submodules and the set of all small submodules of M, respectively. It is clear that the NCS condition is a generalization of the well-known CS condition. Properties of the NCS conditions of modules and rings are explored in this article. In the end, it is proved that a ring R is right Σ-CS if and only if R is right perfect and right countably Σ-NCS. Recall that a ring R is called right Σ-CS if every direct sum of copies of R_R is a CS module. And a ring R is called right countably Σ-NCS if every direct sum of countable copies of R_R is an NCS module.

We compute explicitly the modular derivations for Poisson-Ore extensions and tensor products of Poisson algebras.

The concepts of strongly lifting modules and strongly dual Rickart modules are introduced and their properties are studied and relations between them are given in this paper. It is shown that a strongly lifting module has the strongly summand sum property and the generalized Hopfian property, and a ring R is a strongly regular ring if and only if R_R is a strongly dual Rickart module, if and only if aRis a fully invariant direct summand of R_Rfor every a ∈ R.

The core inverse for a complex matrix was introduced by O. M. Baksalary and G. Trenkler. D. S. Rakíc, N.Č. Diňcíc and D. S. Djordjevíc generalized the core inverse of a complex matrix to the case of an element in a ring. They also proved that the core inverse of an element in a ring can be characterized by five equations and every core invertible element is group invertible. It is natural to ask when a group invertible element is core invertible. In this paper, we will answer this question. Let R be a ring with involution, we will use three equations to characterize the core inverse of an element. That is, let a, b ∈ R. Then a ∈ R^# with a^# = b if and only if (ab)∗ = ab, ba²= a, and ab²= b. Finally, we investigate the additive property of two core invertible elements. Moreover, the formulae of the sum of two core invertible elements are presented.

Let R be a prime ring of characteristic not 2, A be an additive subgroup of R, and F, T, D,K: A → R be additive maps such that F[x, y]) = F(x)y − yK(x) − T(y)x + xD(y) for all x, y ∈ A. Our aim is to deal with this functional identity when A is R itself or a noncentral Lie ideal of R. Eventually, we are able to describe the forms of the mappings F, T, D, and K in case A = R with deg(R)>3 and also in the case A is a noncentral Lie ideal and deg(R)>9. These enable us in return to characterize the forms of both generalized Lie derivations, D-Lie derivations and Lie centralizers of R under some mild assumptions. Finally, we give a generalization of Lie homomorphisms on Lie ideals.