Department of Mathematics
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Let N be an indecomposable totally transcendental module left R module, and let S be the endomorphism ring of N. S is a local ring. It is part of the folk-lore of the subject area that for each ordinal α the α-th elementary socle of N is the annihilator in N of the α-th (left) power of the Jacobson radical of S.

In attempting to write up a proof of this "fact" I came up against a curious block: the expected proof by transfinite induction appears to break down at limit stages, which are usually dismissed as trivial in problems like this.

I will review the basic properties of indecomposable tt modules and the elementary socle series, and work through the proof as far as it goes. Perhaps we will see a way through to the end? The construction of any counterexample will be inherently difficult.