Morita Rings Research Articles

Cotorsion pairs and model structures on Morita rings

We study cotorsion pairs and abelian model structures on Morita rings Λ=(ANBAMABB) which are Artin algebras. Given cotorsion pairs (U,X) and (V,Y) in A-Mod and B-Mod, respectively, one can construct four cotorsion pairs in Λ-Mod:((XY)⊥,(XY)),(Δ(U,V),Δ(U,V)⊥),((UV),(UV)⊥),(∇⊥(X,Y),∇(X,Y)). These cotorsion pairs have relations:Δ(U,V)⊥⊆(XY),∇⊥(X,Y)⊆(UV). An important feature is that they are not equal, in general. In fact, there even exists a Morita algebra Λ, such that the four cotorsion pairs are pairwise different. The problem of identifications, i.e., when these inclusions are the same, are studied; the heredity and completeness of these cotorsion pairs are investigated; and finally, various model structures on Λ-Mod are obtained, by explicitly giving the corresponding Hovey triples and Quillen's homotopy categories. In particular, cofibrantly generated Hovey triples, and the Gillespie-Hovey triples induced by compatible generalized projective (respectively, injective) cotorsion pairs, are explicitly constructed. All these Hovey triples obtained are pairwise different and “new” in some sense. Some results are even new when M=0 or N=0.

Frobenius pairs arising from Morita rings

This paper is to study how to construct Frobenius pairs over a Morita ring [Formula: see text]. Given Frobenius pairs [Formula: see text] and [Formula: see text] in [Formula: see text] and [Formula: see text], respectively, we construct Frobenius pairs [Formula: see text] and [Formula: see text] in [Formula: see text] under some conditions. As an application, we establish a sufficient and necessary condition of Gorenstein projective modules over the Morita ring [Formula: see text].

Silting modules over a class of Morita rings

Abstract Let Δ = A N B A M A B B \Delta =\left(\begin{array}{cc}A& {}_{A}N_{B}\\ {}_{B}M_{A}& B\end{array}\right) be a Morita ring, where M ⊗ A N = 0 = N ⊗ B M M{\otimes }_{A}N=0=N{\otimes }_{B}M . Let X X be left A A -module and Y Y be left B B -module. We prove that ( X , M ⊗ A X , 1 , 0 ) ⊕ ( N ⊗ B Y , Y , 0 , 1 ) \left(X,M{\otimes }_{A}X,1,0)\oplus \left(N{\otimes }_{B}Y,Y,0,1) is a silting module if and only if X X is a silting A A -module, Y Y is a silting B B -module, M ⊗ A X M{\otimes }_{A}X is generated by Y Y , and N ⊗ B Y N{\otimes }_{B}Y is generated by X X . As a consequence, we obtain that if M A {M}_{A} and N B {N}_{B} are flat, then ( X , M ⊗ A X , 1 , 0 ) ⊕ ( N ⊗ B Y , Y , 0 , 1 ) \left(X,M{\otimes }_{A}X,1,0)\oplus \left(N{\otimes }_{B}Y,Y,0,1) is a tilting Δ \Delta -module if and only if X X is a tilting A A -module, Y Y is a tilting B B -module, M ⊗ A X M{\otimes }_{A}X is generated by Y Y , and N ⊗ B Y N{\otimes }_{B}Y is generated by X X .

Strongly Gorenstein-injective modules over Morita rings

In this paper, we consider Morita rings with zero bimodule homomorphisms. We establish necessary and sufficient conditions for all strongly complete injective resolutions over a Morita ring [Formula: see text]. We also obtain necessary and sufficient conditions for all strongly Gorenstein-injective modules over [Formula: see text].

How to construct Gorenstein projective modules relative to complete duality pairs over Morita rings

Let [Formula: see text] be a Morita ring with [Formula: see text]. We first study how to construct (complete) duality pairs of [Formula: see text]-modules using (complete) duality pairs of [Formula: see text]-modules and [Formula: see text]-modules, generalizing the result of Mao (Commun. Algebra 12 (2020) 5296–5310) about the duality pairs over a triangular matrix ring. Moreover, we construct Gorenstein projective modules relative to complete duality pairs of [Formula: see text]-modules. Finally, we give an application to Ding projective modules.

Construction of Gorenstein-projective modules over Morita rings

In this paper, we obtain necessary and sufficient conditions for all complete projective resolutions over a Morita ring [Formula: see text]. As special cases, we get a class of complete projective resolutions over [Formula: see text], from the ones over [Formula: see text] and [Formula: see text]. We also obtain necessary and sufficient conditions for all Gorenstein-projective modules over a Morita ring [Formula: see text]. As special cases, we get a class of Gorenstein-projective modules over [Formula: see text], from the ones over [Formula: see text] and [Formula: see text]. Furthermore, we determine all complete projective resolutions as well as all Gorenstein-projective modules over the [Formula: see text] matrix algebra [Formula: see text] over [Formula: see text].

Igusa–Todorov algebras over Morita rings

Let [Formula: see text] be a Morita ring which is an Artin algebra and has zero bimodule homomorphisms. Assume that [Formula: see text] and [Formula: see text] are projective modules. For any positive integer [Formula: see text], it is proved that [Formula: see text] is [Formula: see text]-Igusa–Todorov (respectively, [Formula: see text]-syzygy-finite) if and only if [Formula: see text] and [Formula: see text] are [Formula: see text]-Igusa–Todorov (respectively, [Formula: see text]-syzygy-finite).

Gorenstein‐Projective Modules over a Class of Morita Rings

Let be a Morita ring such that the bimodule homomorphisms are zero. In this paper, we give sufficient conditions for a Δ(0,0)‐module (X, Y, f, g) to be Gorenstein‐projective. As an application, we give sufficient conditions when the algebras A and B inherit the strongly CM‐freeness of Δ(0,0).

Gorenstein-Projective Modules over Morita Rings

Let [Formula: see text] be a Morita ring which is an Artin algebra. In this paper we investigate the relations between the Gorenstein-projective modules over a Morita ring [Formula: see text] and the algebras [Formula: see text] and [Formula: see text]. We prove that if [Formula: see text] is a Gorenstein algebra and both [Formula: see text] and [Formula: see text] (resp., both [Formula: see text] and [Formula: see text]) have finite projective dimension, then [Formula: see text] (resp., [Formula: see text]) is a Gorenstein algebra. We also discuss when the CM-freeness and the CM-finiteness of a Morita ring [Formula: see text] is inherited by the algebras [Formula: see text] and [Formula: see text].

RSS equivalences over a class of Morita rings

For two bimodules NBA and MAB with M⊗AN=0=N⊗BM, the monomorphism category M(A,M,N,B) and its dual, the epimorphism E(A,M,N,B), are introduced and studied. By definition, M(A,M,N,B) is the subcategory of Δ-mod consisting of (X,Y,f,g) such that f:M⊗AX→Y is a monic B-map and g:N⊗BY→X is a monic A-map, where Δ=(ANMB) is a Morita ring. This monomorphism category is a resolving subcategory of Δ-mod if and only if MA and NB are projective modules; and if this is the case and if in addition HomB(N,B)A and HomA(M,A)B are projective modules, then it has enough relative injective objects, and it is a Frobenius category if and only if A and B are selfinjective. Under these conditions we prove that there is a unique cotilting left Δ-module T, up to the multiplicity of the indecomposable direct summands, such that M(A,M,N,B)=T⊥. When M and N are exchangeable bimodules, Ringel-Schmidmeier-Simson equivalence between M(A,M,N,B) and E(A,M,N,B) are given.

The Extension Dimension of Abelian Categories

Let $\A$ be an abelian category having enough projective objects and enough injective objects. We prove that if $\A$ admits an additive generating object, then the extension dimension and the weak resolution dimension of $\A$ are identical, and they are at most the representation dimension of $\A$ minus two. By using it, for a right Morita ring $\La$, we establish the relation between the extension dimension of the category $\mod \La$ of finitely generated right $\Lambda$-modules and the representation dimension as well as the right global dimension of $\Lambda$. In particular, we give an upper bound for the extension dimension of $\mod \Lambda$ in terms of the projective dimension of certain class of simple right $\Lambda$-modules and the radical layer length of $\Lambda$. In addition, we investigate the behavior of the extension dimension under some ring extensions and recollements.

Morphism Categories of Gorenstein-projective Modules

Let Λ be an algebra of finite Cohen-Macaulay type and Γ its Cohen-Macaulay Auslander algebra. We are going to characterize the morphism category Mor(Λ-Gproj) of Gorenstein-projective Λ-modules in terms of the module category Γ-mod by a categorical equivalence. Based on this, we obtain that some factor category of the epimorphism category Epi(Λ-Gproj) is a Frobenius category, and also, we clarify the relations among Mor(Λ-Gproj), Mor(T2Λ-Gproj) and Mor(Δ-Gproj), where T2(Λ) and Δ are respectively the lower triangular matrix algebra and the Morita ring closely related to Λ.

Comparisons of Left Recollements

Let (F′, F, F″) be a comparison of left recollements of triangulated categories such that F′ and F″ are equivalences. We prove that if F is full then F is an equivalence; and on the other hand, we construct a class of examples via the derived categories of Morita rings, showing that there really exists such a comparison (F′, F, F″) so that F is not an equivalence. This is in contrast to the case of a recollement. We also give a class of examples of left recollements of homotopy categories, which can not sit in recollements.

Gorenstein Homological Aspects of Monomorphism Categories via Morita Rings

For any ring R the category of monomorphisms is a full subcategory of the morphsim category over R, where the latter is equivalent to the module category of the triangular matrix ring with entries the ring R. In this work, we consider the monomorphism category as a full subcategory of the module category over the Morita ring with all entries the ring R and zero bimodule homomorphisms. This approach provides an interesting link between Morita rings and monomorphism categories. The aim of this paper is two-fold. First, we construct Gorenstein-projective modules over Morita rings with zero bimodule homomorphisms and we provide sufficient conditions for such rings to be Gorenstein Artin algebras. This is the first part of our work which is strongly connected with monomorphism categories. In the second part, we investigate monomorphisms where the domain has finite projective dimension. In particular, we show that the latter category is a Gorenstein subcategory of the monomorphism category over a Gorenstein algebra. Finally, we consider the category of coherent functors over the stable category of this Gorenstein subcategory and show that it carries a structure of a Gorenstein abelian category.

On Artin Algebras Arising from Morita Contexts

We study Morita rings \(\Lambda _{(\phi ,\psi )}=\left (\begin {array}{cc}A &_{A}N_{B} _{B}M_{A} & B \end {array}\right )\) in the context of Artin algebras from various perspectives. First we study covariantly finite, contravariantly finite, and functorially finite subcategories of the module category of a Morita ring when the bimodule homomorphisms \(\phi \) and \(\psi \) are zero. Further we give bounds for the global dimension of a Morita ring \(\Lambda _{(0,0)}\), as an Artin algebra, in terms of the global dimensions of A and B in the case when both \(\phi \) and \(\psi \) are zero. We illustrate our bounds with some examples. Finally we investigate when a Morita ring is a Gorenstein Artin algebra and then we determine all the Gorenstein-projective modules over the Morita ring \(\Lambda _{\phi ,\psi }\) in case \(A=N=M=B\) and A an Artin algebra.

DERIVATIONS ON MORITA RINGS AND GENERALIZED DERIVATIONS

In this note, derivations on Morita rings (generalized matrix rings) are introduced and generalized derivations on rings are proved to be Morita invariant without involvement of any homology theory.

Ideals in Morita Rings and Morita Semigroups

We characterize the lattice of all ideals of a Morita ring (semigroup) when the corresponding pair of rings (semigroups) in the Morita context are Morita equivalent s–unital (like–unity) rings (semigroups).

Purity and Almost Split Morphisms in Abstract Homotopy Categories: A Unified Approach via Brown Representability

Our aim in this paper is to develop a theory of purity and to prove in a unified conceptual way the existence of almost split morphisms, almost split sequences and almost split triangles in abstract homotopy categories, a rather omnipresent class of categories of interest in representation theory. Our main tool for doing this is the classical Brown representability theorem. Mathematics Subject Classifications (2000): 16Gxx, 18Gxx, 18Exx, 55U35. Abstract homotopy categories were introduced by E. Brown in the mid-sixties as the proper framework for the study of the homotopy theory of CW-complexes. In this setting, he proved in (18) his celebrated representability theorem, a variant of which has recently found important applications in the stable module category of a modular group algebra and more generally in compactly generated triangulated categories, mainly through the work of Rickard (42) and Neeman (40). Our main purpose in this paper is to develop a theory of purity and a theory of existence of almost split morphisms in an abstract homotopy category, using the Brown representability theorem as a main catalyzing tool. This aim is justified by the fact that abstract homotopy categories are om- nipresent in representation theory. They include module categories, locally finitely presented categories with products, compactly generated triangulated categories, categories of projective modules, and stable categories modulo projectives over left coherent and right perfect rings, categories of injective modules and stable categories modulo injectives over right Morita rings, (stable) categories of Cohen- Macaulay modules over Gorenstein rings, and many others. Abstract homotopy

Morita duality and finitely group-graded rings

We give the relation between the (rigid) graded Morita duality and the Morita duality on a finitely group-graded ring and the relation between a left Morita ring and some of its matrix rings.

Polynomial rings over Jacobson-Hilbert rings

All rings considered are commutative with unit. A ring R is SISI (in Vamos' terminology) if every subdirectly irreducible factor ring R/I is self-injective. SISI rings include Noetherian rings, Morita rings and almost maximal valuation rings ([V1]). In [F3] we raised the question of whether a polynomial ring R[x] over a SISI ring R is again SISI. In this paper we show this is not the case.