Kompaktifikaation

Kompaktifikaation is a concept within abstract algebra, specifically related to ring theory. It describes a process where a commutative ring is embedded into a larger ring in such a way that certain properties are preserved or extended. The most common context for kompaktifikaation is the embedding of a commutative ring $R$ into a ring $R'$ such that $R'$ is a total ring of quotients of $R$. This means that every non-zero divisor in $R$ becomes invertible in $R'$.

A key aspect of kompaktifikaation is that it often aims to create a ring where division by non-zero elements is always possible. This is analogous to how the field of rational numbers $\mathbb{Q}$ is constructed from the ring of integers $\mathbb{Z}$ by formally inverting all non-zero integers. The resulting field $\mathbb{Q}$ is the total ring of quotients of $\mathbb{Z}$.

In a more general setting, kompaktifikaation can refer to constructing a ring that is "universal" in some sense, often involving localization techniques. For example, the construction of the localization of a ring $R$ with respect to a multiplicatively closed set $S$ can be seen as a form of kompaktifikaation, where elements of $S$ are made invertible. The resulting localized ring $S^{-1}R$ retains many of the algebraic properties of the original ring $R$ while allowing for certain divisions. The term itself, while not universally standardized across all algebraic literature, points to the idea of making a ring more "complete" or "compact" with respect to its quotients.