The representations of the Dynkin quivers and the corresponding Euclidean quivers are treated in many books. These notes provide three building blocks for dealing with representations of Dynkin (and Euclidean) quivers. They should be helpful as part of a direct approach to study represen-tations of quivers, and they shed some new light on properties of Dynkin and Euclidean quivers.

In this paper, which is a cont inuation of our previous paper [T. Albu, M. Iosif, A. Tercan, The conditions (C_i) in modular lattices, and applications, J. Algebra Appl. 15 (2016), http: dx.doi.org/10.1142/S0219498816500018], we investigate the latticial counterparts of some results about modules satisfying the conditions (C₁₁) or (C₁₂). Applications are given to Grothendieck categories and module categories equipped with hereditary torsion theories.

Let D be an integral domain, V(D) (resp., t-V(D)) be the set of all valuation (resp., t-valuation) ideals of D, and w-P(D) be the set of primary w-ideals of D. Let D[X] be the polynomial ring over D, c(f) be the ideal of D generated by the coefficients of f ∈ D[X], and N_v= {f ∈ D[X] | c(f)_v=D}. In this paper, we study integral domains D in which w-P(D) ⊆ t-V(D), t-V(D) ⊆ w-P(D), or t-V(D) = w-P(D). We also study the relationship between t-V(D) and V(D[X]Nv), and characterize when t-V(A + XB[X]) ⊆w-P(A + XB[X]) holds for a proper extension A ⊂ B of integral domains.

We investigate the comodule representation category over the Morita-Takeuchi context coalgebra Γ and study the Gorensteinness of Γ. Moreover, we determine explicitly all Gorenstein injective comodules over the Morita-Takeuchi context coalgebra Γ and discuss the localization in Gorenstein coalgebras. In particular, we describe its Gabriel quiver and carry out some examples when the Morita-Takeuchi context coalgebra is basic.

We study structures of Hochschild 2-cocycles related to endomorphisms and introduce a skew Hochschild 2-cocycle. We moreover define skew Hochschild extensions equipped with skew Hochschild 2-cocycles, and then we examine uniquely clean, Abelian, directly finite, symmetric, and reversible ring properties of skew Hochschild extensions and related ring systems. The results obtained here provide various kinds of examples of such rings. Especially, we give an answer negatively to the question of H. Lin and C. Xi for the corresponding Hochschild extensions of reversible (or semicommutative) rings. Finally, we establish three kinds of Hochschild extensions with Hochschild 2-cocycles and skew Hochschild 2-cocycles.

We study complete cohomology of complexes with finite Gorenstein AC-projective dimension. We show first that the class of complexes admitting a complete level resolution is exactly the class of complexes with finite Gorenstein AC-projective dimension. This lets us give some general techniques for computing complete cohomology of complexes with finite Gorenstein ACprojective dimension. As a consequence, the classical relative cohomology for modules of finite Gorenstein AC-projective dimension is extended. Finally, the relationships between projective dimension and Gorenstein AC-projective dimension for complexes are given.

Let H₂ be Sweedler’s 4-dimensional Hopf algebra and r(H₂) be the corresponding Green ring of H₂. In this paper, we investigate the automorphism groups of Green ring r(H₂) and Green algebra F(H₂) = r(H₂) ⊗FZ, where F is a field, whose characteristics is not equal to 2. We prove that the automorphism group of r(H₂) is isomorphic to K₄, where K₄ is the Klein group, and the automorphism group of F(H₂) is the semidirect product of Z₂ and G, where G= F \ {1/2} with multiplication given by a · b= 1− a − b+ 2ab.

Is a semiprimary right self-injective ring a quasi-Frobenius ring? Almost half century has passed since Faith raised this problem. He first conjectured “No” in his book Algebra II. Ring Theory in 1976, but changing his mind, he conjectured “Yes” in his article “When self-injective rings are QF: a report on a problem” in 1990. In this paper, we describe recent studies of this problem based on authors works and raise related problems.

We obtain the structure of the rings in which every element is either a sum or a difference of a nilpotent and an idempotent that commute. This extends the structure theorems of a commutative weakly nil-clean ring, of an abelian weakly nil-clean ring, and of a strongly nil-clean ring. As applications, this result is used to determine the 2-primal rings R such that the matrix ring Mn(R) is weakly nil-clean, and to show that the endomorphism ring End_D(V ) over a vector space V_D is weakly nil-clean if and only if it is nil-clean or dim(V ) = 1 with D≅Z3.

We give sufficient conditions and necessary conditions on duality preservability of Auslander-Reiten quivers of derived categories and cluster categories over hereditary algebras. Meantime, we characterize the condition of generalized path algebras as cleft extensions of path algebras.

Given any integers a, b, c, and d with a>1, c≥0, b≥a + c, and d≥b + c, the notion of (a, b, c, d)-Koszul algebra is introduced, which is another class of standard graded algebras with “nonpure” resolutions, and includes many Artin-Schelter regular algebras of low global dimension as specific examples. Some basic properties of (a, b, c, d)-Koszul algebras/modules are given, and several criteria for a standard graded algebra to be (a, b, c, d)- Koszul are provided.

We consider a one point extension algebra B of a quiver algebra Aq over a field k defined by two cycles and a quantum-like relation depending on a nonzero element q in k. We determine the Hochschild cohomology ring of B modulo nilpotence and show that if q is a root of unity, then B is a counterexample to Snashall-Solberg’s conjecture.

Let Mbe a monoid. A ring Ris called M-π-Armendariz if whenever α = a₁g₁+ a₂g₂+ · · · + a_ng_n, β = b₁h₁+ b₂h₂+ · · · + b_mh_m ∈ R[M] satisfy αβ ∈ nil(R[M]), then aibj ∈ nil(R) for all i, j. A ring R is called weakly 2-primal if the set of nilpotent elements in R coincides with its Levitzki radical. In this paper, we consider some extensions of M-π-Armendariz rings and further investigate their properties under the condition that R is weakly 2-primal. We prove that if R is an M-π-Armendariz ring then nil(R[M]) = nil(R)[M]. Moreover, we study the relationship between the weak zip-property (resp., weak APP-property, nilpotent p.p.-property, weak associated prime property) of a ring R and that of the monoid ring R[M] in case R is M-π-Armendariz.

For a square-free integer d other than 0 and 1, let K=?(d), where ? is the set of rational numbers. Then K is called a quadratic field and it has degree 2 over ?. For several quadratic fields K=?(d), the ring R_dof integers of K is not a unique-factorization domain. For d<0, there exist only a finite number of complex quadratic fields, whose ring Rd of integers, called complex quadratic ring, is a unique-factorization domain, i.e., d = −1,−2,−3,−7,−11,−19,−43,−67,−163. Let ϑ denote a prime element of Rd, and let n be an arbitrary positive integer. The unit groups of Rd/⟨vn⟩ was determined by Cross in 1983 for the case d = −1. This paper completely determined the unit groups of Rd/⟨vn⟩ for the cases d = −2,−3.

Motivated by τ-tilting theory developed by T. Adachi, O. Iyama, I. Reiten, for a nite-dimensional algebra Λwith action by a nite group G; we introduce the notion of G-stable support τ-tilting modules. Then we establish bijections among G-stable support τ-tilting modules over Λ; G-stable two-term silting complexes in the homotopy category of bounded complexes of nitely generated projective Λ-modules, and G-stable functorially nite torsion classes in the category of nitely generated left Λ-modules. In the case when Λ is the endomorphism of a G-stable cluster-tilting object T over a Hom-nite 2-Calabi-Yau triangulated category ℓ with a G-action, these are also in bijection with G-stable cluster-tilting objects in ℓ : Moreover, we investigate the relationship between stable support τ-tilitng modules over Λ and the skew group algebra ΛG:

We introduce two adjoint pairs ({{\displaystyle e}}_{\lambda }^i,{{\displaystyle ( )}}_i) and({{\displaystyle ( )}}_i,{{\displaystyle e}}_{\rho }^i) and give a new method to construct cotorsion pairs. As applications, we characterize all projective and injective representations of a generalized path algebra and exhibit projective and injective objects of the category{{\displaystyle \mathfrak{M}}}_p which is a generalization of monomorphisms category.