Pure semisimplicity conjecture and Artin problem for dimension sequences

Inspired by a recent paper due to José Luis García, we revisit the attempt of Daniel Simson construct counterexample pure semisimplicity conjecture. Using compactness, show that existence such would readily follow from very certain (countable set of) hereditary artinian rings finite representation type. The is then proved be equivalent special types embeddings, which call tight, division into simple rings. tools Aidan Schofield 1980s, can an embedding F↪Mn(G) exists provided n<5. As byproduct, obtain ring extension G⊆F bimodule FFG has right dimension sequence (1,2,2,2,1,4). Finally, formulate Conjecture A, asserts particular type adjunction element made, and demonstrate its validity sufficient prove tight embeddings in general, hence disprove