Lia Vas
We prove that every perfect torsion theory for a ring $R$ is differential (in the sense of [P. E. Bland, Differential torsion theory, Journal of Pure and Applied Algebra 204 (2006) 1 -- 8]). In this case, we construct the extension of a derivation of a right $R$-module $M$ to a derivation of the module of quotients of $M$. Then, we prove that the Lambek and Goldie torsion theories for any $R$ are differential.
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A torsion theoretical characterization of left Noetherian rings $R$ over which injective hulls of simple left modules are locally Artinian is given. Sufficient conditions for a left Noetherian ring to satisfy this finiteness condition are obtained in terms of torsion theories.
A torsion theory is called differential (higher differential) if a derivation (higher derivation) can be extended from any module to the module of quotients corresponding to the torsion theory. We study conditions equivalent to higher differentiability of a torsion theory. It is known that the Lambek, Goldie and any perfect torsion theories are differential. We show that these classes of torsion theories are higher differential as well. Then, we study conditions under which a higher derivation extended to a right module of quotients extends also to a right module of quotients with respect to a…
We introduce a linearly ordered lattice $μ(Grp)$ of torsion theories in simplicial groups. The torsion theories are defined where the torsion/torsion-free subcategories are given by the simplicial groups with bounded above/below Moore complex, respectively. These torsion theories extend naturally the torsion theories in internal groupoids in groups. Connections of this lattice with the homotopy groups are established since the homotopy groups of a simplicial group can be calculated as the quotients of torsion subojects.
The study of modules over a finite von Neumann algebra ${\mathcal A}$ can be advanced by the use of torsion theories. In this work, some torsion theories for ${\mathcal A}$ are presented, compared and studied. In particular, we prove that the torsion theory $(\mathrm{\bf T},\mathrm{\bf P})$ (in which a module is torsion if it is zero-dimensional) is equal to both Lambek and Goldie torsion theories for ${\mathcal A}$. Using torsion theories, we describe the injective envelope of a finitely generated projective ${\mathcal A}$-module and the inverse of the isomorphism $K_0({\mathcal A})\to K_0({\…
The dimension of any module over an algebra of affiliated operators ${\mathcal U}$ of a finite von Neumann algebra ${\mathcal A}$ is defined using a trace on ${\mathcal A}.$ All zero-dimensional ${\mathcal U}$-modules constitute the torsion class of torsion theory $(\mathrm{\bf T},\mathrm{\bf P})$. We show that every finitely generated ${\mathcal U}$-module splits as the direct sum of torsion and torsion-free part. Moreover, we prove that the theory $(\mathrm{\bf T},\mathrm{\bf P})$ coincides with the theory of bounded and unbounded modules and also with the Lambek and Goldie torsion theories.…
First, we show that a compact object $C$ in a triangulated category, which satisfies suitable conditions, induces a $t$-structure. Second, in an abelian category we show that a complex $P^{\centerdot}$ of small projective objects of term length two, which satisfies suitable conditions, induces a torsion theory. In the case of module categories, using a torsion theory, we give equivalent conditions for $P^{\centerdot}$ to be a tilting complex. Finally, in the case of artin algebras, we give one to one correspondence between tilting complexes of term length two and torsion theories with certain …
Many known results on finite von Neumann algebras are generalized, by purely algebraic proofs, to a certain class ${\mathcal C}$ of finite Baer *-rings. The results in this paper can also be viewed as a study of the properties of Baer *-rings in the class ${\mathcal C}$. First, we show that a finitely generated module over a ring from the class ${\mathcal C}$ splits as a direct sum of a finitely generated projective module and a certain torsion module. Then, we define the dimension of any module over a ring from ${\mathcal C}$ and prove that this dimension has all the nice properties of the di…
Torsion aspects of varieties of simplicial groups
该研究探讨单纯群的挠理论格，分析其在特定子范畴的限制，证明2-交叉模和群交叉复形范畴的半阿贝尔性并描述其挠理论，还得到新的预挠理论与TTF理论实例。
We extend some recent results on the differentiability of torsion theories. In particular, we generalize the concept of $(α, β)$-derivation to $(α, β)$-higher derivation and demonstrate that a filter of a hereditary torsion theory that is invariant for $α$ and $β$ is $(α, β)$-higher derivation invariant. As a consequence, any higher derivation can be extended from a module to its module of quotients. Then, we show that any higher derivation extended to a module of quotients extends also to a module of quotients with respect to a larger torsion theory in such a way that these extensions agree. …
In this article we explore a non-abelian torsion theory in the category of preordered groups: the objects of its torsion-free subcategory are the partially ordered groups, whereas the objects of the torsion subcategory are groups (with the total order). The reflector from the category of preordered groups to this torsion-free subcategory has stable units, and we prove that it induces a monotone-light factorization system. We describe the coverings relative to the Galois structure naturally associated with this reflector, and explain how these coverings can be classified as internal actions of …
Let R be a ring and let T be a hereditary torsion class of R-modules. The inclusion of the localizing subcategory generated by T into the derived category of R has a right adjoint, which is a colocalization. Benson has recently shown how to compute this right sdjoint when R is the group ring of a finite group over a prime field and T is the hereditary torsion class generated by a simple module. We generalize Benson's construction to the case where T is any hereditary torsion class on R. This yields an explicit formula for the colocalization with respect to T, using an injective cogenerator for…
We begin a study of torsion theories for representations of an important class of associative algebras over a field which includes all finite W-algebras of type A, in particular the universal enveloping algebra of gl(n) (or sl(n)) for all n. If U is such and algebra which contains a finitely generated commutative subalgebra A, then we show that any A-torsion theory defined by the coheight of prime ideals is liftable to U. Moreover, for any simple U-module M, all associated prime ideals of M in Spec A have the same coheight. Hence,thecoheight of the associated prime ideals of A is an invariant …
This is the second of two papers on the injective spectrum of a right noetherian ring. In the prequel, we considered the injective spectrum as a topological space associated to a ring (or, more generally, a Grothendieck category), which generalises the Zariski spectrum. We established some results about the topology and its links with Krull dimension, and computed a number of examples. In the present paper, which can largely be read independently of the first, we extend these results by defining a sheaf of rings on the injective spectrum and considering sheaves of modules over this structure s…
In abelian categories like the category of R-modules and even in the category S-Act 0 of S-acts with a unique zero, idempotent radicals and torsion theories are equivalent, and the τ-torsion and τ- torsion free classes of a torsion theory τ are closed under coproducts. These are not necessarily true in the category S-Act of S-acts. In this paper, we prove that torsion theories are equivalent with the Kurosh- Amitsur radicals. We, also, show that the class of Kurosh-Amitsur radicals is a reflective subcategory of Hoehnke radicals, as a poset.
Lattices of pretorsion classes
本文研究了mod A中pretorsion类的格L_t(A)，探讨了其与torsion类格的关系，描述了完全join-irreducible元素，并给出了L_t(A)为分布的条件及与分布闭包的识别条件。
In the present notes we generalize the classical work of Demazure [Invariants symétriques entiers des groupes de Weyl et torsion] to arbitrary oriented cohomology theories and formal group laws. Let G be a split semisemiple linear algebraic group over a field and let T be its split maximal torus. We construct a generalized characteristic map relating the so called formal group ring of the character group of T with the cohomology of the variety of Borel subgroups of G. The main result of the paper says that the kernel of this map is generated by W-invariant elements, where W is the Weyl group o…
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