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The Transfer of the Krull Dimension and the Gabriel Dimension to Subidealizers

Let M be a right ideal of the ring T with identity. A unital subring R of T which contains M as a two-sided ideal is called a subidealizer ; the largest such subring is the idealizer I (M) of M in T. M is said to be generative if TM = T. In this case M is idempotent, and it follows from the dual basis lemma that T is finitely generated projective as a right R-module (see [7, Lemma 2.1]); we make frequent use of these two facts in this paper.