Normalisent

Normalisent is a term used in mathematics, particularly in abstract algebra and algebraic geometry, to describe a process of simplifying or "cleaning up" a ring or an algebraic variety. It aims to remove certain undesirable features, such as nilpotents or zero-divisors, that can complicate structural analysis.

In the context of commutative rings, the normalisation of a ring R, denoted by N(R), is the subring of the integral closure of R in its total ring of fractions that consists of elements integral over R. This process essentially removes nilpotent elements from R. An element 'x' is nilpotent if there exists a positive integer 'n' such that x^n = 0. The normalisation of a ring is always a reduced ring, meaning it has no non-zero nilpotent elements.

For algebraic varieties, the normalisation refers to replacing a singular variety with a non-singular one that is birational to the original. This means there is a map between the varieties that is an isomorphism except possibly on a set of measure zero. The normalisation process aims to resolve singularities, making the variety geometrically simpler and easier to study. The normalisation of an irreducible algebraic variety is always an irreducible and normal variety. A variety is called normal if its local rings are all normal domains.

The concept of normalisation is crucial in various areas of mathematics, including commutative algebra, algebraic geometry, and number theory, as it provides a standard form for rings and varieties, facilitating deeper understanding and analysis.