lokalringen

The term "lokalringen" is a Swedish word that translates to "local ring" in English. In mathematics, specifically in abstract algebra, a local ring is a commutative ring with unity that has a unique maximal ideal. This unique maximal ideal is sometimes referred to as the "Jacobson radical" of the ring.

A key property of a local ring is that if the sum of elements from a set is a unit, then at least one of the elements in that set must be a unit. This property is crucial in various areas of algebra, including algebraic geometry and number theory. For instance, the ring of integers localized at a prime number $p$, denoted $\mathbb{Z}_{(p)}$, is a common example of a local ring. In this ring, all fractions $a/b$ where $p$ does not divide $b$ are included, and the unique maximal ideal consists of fractions where $p$ divides $a$.

Local rings play a significant role in studying the local properties of schemes and varieties in algebraic geometry. They allow mathematicians to analyze the behavior of geometric objects at specific points by examining the algebraic structure of the corresponding local rings. The concept is fundamental for understanding singularities and other local phenomena in geometry.