Noncommutative Krull dimension (${\mathrm{Kdim}}_{R}M$)

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.