Kiertymämoduuleita

Kiertymämoduuleita, known in English as twist modules or torsion modules, are a concept in abstract algebra, specifically within the theory of modules over a ring. A kiertymämoduuli is a module that possesses torsion elements. A torsion element in a module $M$ over a ring $R$ is an element $m \in M$ for which there exists a non-zero element $r \in R$ such that $r \cdot m = 0$. The element $r$ is often referred to as an annihilator of $m$.

Modules that do not contain any non-zero torsion elements are called torsion-free modules. The concept of torsion is particularly important when dealing with modules over integral domains, where the existence of non-zero divisors is guaranteed. In such cases, if $r \cdot m = 0$ and $r \neq 0$, then $m$ must be zero if $M$ is torsion-free.

The set of all torsion elements in a module $M$ forms a submodule, known as the torsion submodule of $M$. This submodule is often denoted by $T(M)$. The quotient module $M/T(M)$ is then a torsion-free module. This decomposition of a module into its torsion submodule and its torsion-free quotient is a fundamental result in module theory, allowing for a deeper understanding of the module's structure. The study of kiertymämoduuleita and torsion-free modules is crucial in various areas of algebra, including representation theory and algebraic geometry.