Department of Mathematics
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The family of indecomposable pure-injective modules over Z is easy to describe, as is the family of indecomposable pure-injective modules over Q[x]. The situation over Z[x] is vastly different: this ring is "wild", and no classification of the (uncountable) family of pure-injective indecomposables is possible. On the other hand, every indecomposable pure-injective Z[x] module is built up out of indecomposable pure-injective Z modules.

Over the last couple of years I have been exploring how this "blow-up" happens; and I will present some of the ways complicated structure can arise.