Category Of Right R-modules Research Articles

Skew-monoidal categories and bialgebroids

Skew-monoidal categories arise when the associator and the left and right units of a monoidal category are, in a specific way, not invertible. We prove that the closed skew-monoidal structures on the category of right R-modules are precisely the right bialgebroids over the ring R. These skew-monoidal structures induce quotient skew-monoidal structures on the category of R–R-bimodules and this leads to the following generalization: Opmonoidal monads on a monoidal category correspond to skew-monoidal structures with the same unit object which are compatible with the ordinary monoidal structure by means of a natural distributive law. Pursuing a Theorem of Day and Street we also discuss monoidal lax comonads to describe the comodule categories of bialgebroids beyond the flat case.

On the heart associated with a torsion pair

Given an associative ring R and a torsion pair (T,F) in the category of right R-modules, the heartH(T,F) associated with (T,F) is an abelian subcategory of the bounded derived category Db(R). In the present paper we deal with the problem of when H(T,F) is equivalent to a category of modules. We show that if (T,F) is a faithful torsion pair, or, in the general case, if R is right poised and semiperfect, then H(T,F) is equivalent to a category of modules if and only if T and F are naturally associated with a complex in Db(R) of length two with finitely generated projective terms.

Cotorsion Theories Cogenerated by a Torsionfree Class

Let R be a right perfect ring, and let (ℱ, 𝒞) be a cotorsion theory in the category of right R-modules ℳ R . In this article, it is shown that every right R-module has a superfluous ℱ-cover if and only if there exists a torsion theory (𝒜, ℬ) such that (ℱ, 𝒞) is cogenerated by ℬ. It is also proved that if (𝒜, ℬ) is a cosplitting torsion theory, then (⊥ℬ, (⊥ℬ)⊥) is a hereditary and complete cotorsion theory, and if (𝒜, ℬ) is a centrally splitting torsion theory, then (⊥ℬ, (⊥ℬ)⊥) is a hereditary and perfect cotorsion theory.

Auslander–Reiten components over pure-semisimple hereditary rings

Let R be a hereditary, indecomposable, left pure-semisimple ring. We show that R has finite representation type if and only if a certain finitely presented module is endofinite, namely, the tilting and cotilting module W studied in L. Angeleri Hügel (2007) [2]. We then apply the tilting and the cotilting functors to study the endomorphism ring of W and its Auslander–Reiten components. Finally, we transfer this information to the category of right R-modules.

Compact objects in stable module categories

Let R be a perfect ring, \(\underline{\tt{Mod}}R\) the stable module category of right R-modules. We show that any compact object in \(\underline{\tt{Mod}}R\) is isomorphic to some finitely generated R-module. Moreover, we apply the above to stable equivalences between module categories.

Relatively Endocoherent Modules

Let MR be a right R-module over a ring R with S = End (MR). We study the coherence of the left S-module SM relative to a torsion theory for the category of right R-modules. Various results are developed, many extending known results.

Relative Flatness, Mittag–Leffler Modules, and Endocoherence

Let M R be a right R-module over a ring R with S = End(M R ). We study the coherence of the left S-module S M relative to a hereditary torsion theory for the category of right R-modules. Various results are developed, many extending known results.

Primeness Conditions on the Lattice of Hereditary Pretorsion Classes#

A lattice ordered monoid is a structure ⟨L;⊕, 0 L ; ≤⟩ where ⟨L;⊕, 0 L ⟩ is a monoid, ⟨L; ≤ ⟩ is a lattice and the binary operation ⊕ distributes over finite meets. If R is an arbitrary ring with identity, then the set ptors R of all hereditary pretorsion classes in the category of right R-modules is an example of a lattice ordered monoid. It is known that right strongly prime and right strongly semiprime rings are characterizable by means of first order sentences in the language of ptors R. A notion of weak primeness and weak semiprimeness is introduced. These are defined by means of first order sentences in ptors R. It is shown that in the presence of a weak commutativity condition, which is also defined in the language of ptors R, the notions of weak primeness [resp. weak semiprimeness] and strong primeness [resp. strong semiprimeness] coincide. It is also proved that within the class of commutative rings, Quasi–Frobenius rings are characterizable by means of a simple sentence in ptors R.

Tensor product and theories of modules

Modules over a ring R, when tensored with an (R, S)-bimodule, are converted to S-modules. Here I consider, from the standpoint of the model theory of modules, the effect of this operation. The primary motivation arises from questions concerning representation type of algebras and interpretability of modules, where such tensor functors play a key role, but this paper is devoted to more general considerations. For instance, the elementary duality of [2] and [1] is generalised here. It is also shown that, although tensor product does not preserve elementary equivalence, one can define the tensor product of two complete theories of R-modules. The results in Section 1 grew out of a number of discussions with T. Kucera.This is our generic situation. We have an (R, S)-bimodule B and we are considering the functor –⊗RBS from Mod – R (the category of right R-modules) to Mod – S which is given on objects by MR ↦ (M ⊗RB)s and has the obvious action on morphisms. There is a somewhat more general situation: namely we may consider the effect of tensoring (T, R)-bimodules over R with B to obtain (T, S)-bimodules. For the intended applications this case is not needed. Moreover, although some results extend to this more general case, there are some which definitely do not (see the example after 2.1). Therefore we confine ourselves to the case first described. We will also consider B itself as a variable and so ask “What is the effect of tensoring R-modules with (R, S)-bimodules?”.

Local bijective gabriel correspondence and relative fully boundedness in τ-noetherian rings

Let Rbe a right τ-noetherian ring, where τ denotes a hereditary torsion theory on the category of right R-modules. It is shown that every essential τ-closed right ideal of every prime homomorphic image of Rcontains a nonzero two-sided ideal if and only if any two τ-torsionfree injective indecomposable right R-modules with identical associated prime ideals are isomorphic, and for any τ-closed prime ideal Pthe annhilator of a finitely generated P-tame right R-module cannot be a prime ideal properly contained in P. Furthermore, if in the last condition finitely generated is replaced by r-noetherian, then all τ-noetherian τ-torsionfree modules turn out to be finitely annihilated.

A Decomposition of Rings Generated by Faithful Cyclic Modules

AbstractA ring R is said to be generated by faithful right cyclics (right finitely pseudo-Frobenius), denoted by GFC (FPF), if every faithful cyclic (finitely generated) right R-module generates the category of right R-modules. The class of right GFC rings includes right FPF rings, commutative rings (thus every ring has a GFC subring - its center), strongly regular rings, and continuous regular rings of bounded index. Our main results are: (1) a decomposition of a semi-prime quasi-Baer right GFC ring (e.g., a semiprime right FPF ring) is achieved by considering the set of nilpotent elements and the centrality of idempotnents; (2) a generalization of S. Page's decomposition theorem for a right FPF ring.

Some properties of purely simple Kronecker modules, I

Let K be an algebraically closed field. A K 2-system is a pair of K-vector spaces ( V, W) together with a K-bilinear map from K 2× V to W. The category of systems is equivalent to the category of right modules over some K-algebra, R. Most of the concepts in the theory of modules over the polynomial ring K[ξ] have analogues in Mod- R. Unlike the purely simple K[ξ]-modules, which are easily described, purely simple R-modules are quite complex. If M is a purely simple R-module of finite rank n then any submodule of M of rank less than n is finite-dimensional. The following corollaries are derived from this fact: 1. 1.|Every non-zero endomorphism of M is monic. 2. 2.|Every torsion-free quotient of M is purely simple. 3. 3.|An ascending union of purely simple R-modules of increasing rank is not purely simple. It is also shown that a large class of torsion-free rank one modules can occur as the quotient of a purely simple system of rank n, n any positive integer. Moreover, starting from a purely simple system another purely simple module M' of the same rank is constructed and M' is shown to be both a submodule of M and a submodule of a rank 1 torsion-free system. Since the category of right R-modules is a full subcategory of right S-modules, where S is any finite-dimensional hereditary algebra of tame type, the paper provides a way of constructing infinite-dimensional indecomposable S-modules.

Direct sums and direct products of canonical pencils of matrices

A theorem of Kulikov characterizes the K[ x]-modules which are direct sums of finite-dimensional (as a K-vector space) indecomposable modules, where K[ x] is the polynomial ring over the field K. In this paper an analogous characterization is given for modules over the ring R, arising from pairs of linear transformations between a pair of complex vector spaces, ( V, W). R is a certain subring of the ring of 3 × 3 complex matrices. The equivalence between the category of right R-modules and the category of systems enables one to work entirely in the category of systems. (A pair of complex vector spaces is a system if and only if there is a C -bilinear map from C 2 × V to W). R-modules that are direct sums of finite-dimensional indecomposable subsystems are called pure-projective. The above characterization of pure-projective R-modules is used to prove that an R-module M is projective if and only if Ext( M, R) = 0. Direct products of finite-dimensional indecomposable R-modules are also studied, and a theorem pinpoints those that are pure-projective. An example of an R-module M that is not pure-projective, but with the property that every finite subset of M is contained in a pure-projective direct summand of M, is given. A by-product of this example is a class of matrices that generalizes the Vandermonde matrices.

Hermitian forms and higher algebraic 𝐾-theory

Introduction. This note promulgates certain invariants for Hermitian forms over a ring with antistructure which are defined utilizing the algebraic ^-theory of the ring in a Z2-equivariant fashion. Low degree cases of these invariants extend the classical rank, discriminant, and Hasse-Witt invariants. The principal result described here is that these invariants (using all degrees) yield the stable classification of forms over many fields and algebras for which low degree cases alone do not usually suffice (because of signature invariants that may be defined). The manner in which this is accomplished is quite amusing, in that it sheds light on why the reductions of the signatures modulo certain powers of 2 are in fact determined by the low degree invariants. More general results can be obtained, e.g. for higher unitary algebraic A'-theory. However, these are suppressed here in favor of an outline of the equivariant algebraic Ktheory upon which this work so crucially relies. This will be dealt with at length in [Gi2], and arithmetic aspects will be pursued further elsewhere. 1. Hermitian forms. If R is a ring and MR is the category of right R-modules, then PR will denote the full, exact subcategory of M# determined by the finitely generated projective modules, and KnR, n E Z, will denote the (GerstenWagoner) algebraic K-theory groups of R (cf. [Ge] ). Recall that an antistructure (a, u) on R consists of an antiautomorphism r »—* r of R and a unit uGR such that uu = 1 and r* = unT, rER. An Hermitian form over (R, a, w), or an (a, u)-form over R, is a pair (P, A) with P E ?R and X: P x P —+ R such that \(x, y 4zf) = X(x, y) + X(x, z)r, \(y, x) = \(x, y)u for xf y, z E P, r E R, and (Ad X): P —> Da? is an isomorphism where (Ad XX*)00 = X(JC, y), x, y E R. Here, Da: VR —* VR is the a-conjugate duality cofunctor Dal = Hom/2(?, R) where HomR(P, R) has right R-module structure (f° r)(x) = r /(x), x E P, r E R, ƒ E Z)aP. There are evident notions of isometry and orthogonal direct sum of (a, w)-forms over R (cf. [Wa] ), resulting in a permutative category U(R, a, u) and (Gersten-Wagoner type) unitary algebraic K-theory groups KUn(R, a, w), n E Z, of (R, a, w). Of primary interest here is KU0(R, a, u) which is just the Grothendieck group of U(R, a, u).

The Cohomological Dimension of a Directed Set

Let R be a ring with identity, and let C be a small, nonempty category. We denote the category of right R-modules by AbR and the category of contravariant functors C → AbR by AbRC*. The limit functoris left exact, and its kth right derived functor is denoted by colimk. The R-cohomological dimension of C is defined byIf there is a unitary ring homomorphism R→S, then it is not difficult to show that cdsC ≦ cdRC.

A Remark on Flat and Projective Modules

It is the purpose of this note to give some characterizations of flat and projective modules, partly in ideal theoretical terms, partly in terms of the exterior product of a module (“puissance extérieure“); cf. (1).We shall consider left modules over a ring R with identity element and without proper zero divisors. The left module M is called flat if X ⊗R M is an exact functor on the category of right R-modules X. If M is flat over a commutative domain R, M is necessarily torsion-free. Therefore when looking for flatness of a module M over a commutative domain, one may assume from the start that M is torsion-free.In the following theorem, we shall not restrict ourselves to commutative rings R, but the modules concerned have to be torsion-free, which, of course, should mean that rm = 0 implies r = 0 or m = 0.