= Noncommutative Krull dimension
{title2=$\operatorname{Kdim}_RM$}
= Gabriel-Rentschler Krull dimension
{c}
{synonym}
Set the dimension of the zero module to $-1$. A nonzero module has dimension zero exactly when it is an <Artinian module>. Inductively, dimension at most $d$ means that in every descending chain the successive quotients eventually have dimension less than $d$. The least such ordinal is the Krull dimension; the ordinal definition also covers dimensions beyond finite integers. The left dimension of a ring is that of its left regular <module>. In noncommutative rings this is not defined simply as the length of a chain of two-sided prime ideals.
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