Torsievering

Torsievering is a term used in some mathematical discussions to denote a framework that seeks to blend torsion phenomena with sieve methods in algebraic structures. The name combines torsion, the study of elements annihilated by ring elements, with sieve methods that filter objects according to prescribed properties. At present there is no universally accepted definition, and the term appears chiefly in informal or exploratory writings. In a schematic sense, a torsievering would attach to each object a filtered collection of its torsion subobjects defined by a sieve-like rule, such that the selected torsion pieces satisfy a prescribed family of conditions, for example respecting a chosen set of ideals or divisibility constraints.

Possible formalizations are speculative. One approach envisions a torsieve as a family of subobjects indexed by a sieve on a category, producing a torsion filtration that interacts compatibly with exact sequences. Another view treats torsievering as a functorial operation that assigns to each object a torsion component refined by sieve data, analogous to how localization or radical operations refine structures.

In relation to established theory, torsievering sits near torsion theory, sieve theory, and localization methods in homological algebra, but remains an emergent idea without a standard framework. Illustrative examples are limited; for a module M over a ring R, the ordinary torsion submodule Tor(M) could be further stratified by a chosen family of ideals, yielding a multi-level torsion filtration.

See also: torsion, sieve theory, torsion theory, localization.