Ringlängen

Ringlängen refers to the concept of ring length in mathematics, specifically within abstract algebra. It is a fundamental property of rings that describes the size of the ring in terms of its additive group structure. A ring is a set equipped with two binary operations, addition and multiplication, satisfying certain axioms. The ring length is defined for commutative rings with unity.

The ring length of a commutative ring R is the maximum length of a chain of subrings R0 ⊂ R1 ⊂ ... ⊂ Rn where each Ri is a proper subring of Ri+1. This means that Ri is contained within Ri+1, but Ri is not equal to Ri+1. The length of such a chain is n. If no such infinite chain exists, the ring is said to have finite ring length.

Rings with finite ring length are important in various areas of algebra. For instance, Noetherian rings and Artinian rings are related to the concept of ring length. A ring is Artinian if it satisfies the descending chain condition on ideals, and Noetherian if it satisfies the ascending chain condition on ideals. For commutative rings with unity, being Artinian implies having finite ring length.

The ring length is also related to the Krull dimension of a ring. The Krull dimension is defined as the supremum of lengths of chains of prime ideals. For Noetherian rings, the ring length can be greater than the Krull dimension, but there are relationships between these concepts. Understanding ring length provides insights into the structure and complexity of algebraic objects like rings.