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Inspired by a recent paper due to Jos\'{e} Luis Garc\'{i}a, we revisit theattempt of Daniel Simson to construct a counterexample to the puresemisimplicity conjecture. Using compactness, we show that the existence ofsuch counterexample would readily follow from the very existence of certain(countable set of) hereditary artinian rings of finite representation type.  The existence of such rings is then proved to be equivalent to the existenceof special types of embeddings, which we call tight, of division rings intosimple artinian rings. Using the tools by Aidan Schofield from 1980s, we canshow that such an embedding $F\hookrightarrow M_n(G)$ exists provided that$n<5$. As a byproduct, we obtain a division ring extension $G\subseteq F$ suchthat the bimodule ${}_GF_F$ has the right dimension sequence $(1,2,2,2,1,4)$.  Finally, we formulate Conjecture A, which asserts that a particular type ofadjunction of an element to a division ring can be made, and demonstrate thatits validity would be sufficient to prove the existence of tight embeddings ingeneral, and hence to disprove the pure semisimplicity conjecture.

Pure semisimplicity conjecture and Artin problem for dimension sequences