ABSTRACT
One of the main aims of this survey article is to show that the pure semisimplicity conjecture (pss R ) stated below has a positive solution for all rings R if and only if for any pair of division rings F, G and any simple F-G-bimodule fMG such that dim MG is finite and dim F M = ∞ one can construct an indecomposable right module of infinite length over the hereditary ring R M = ( F F M G 0 G ) https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780367810603/b7b45e7e-db6c-4882-a65d-08c75e14529e/content/inequ18_345_1.tif"/> , or equivalently, one can construct a sequence X 1 → f 1 X 2 → ⋯ → X m → f m X m + 1 → ⋯ https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780367810603/b7b45e7e-db6c-4882-a65d-08c75e14529e/content/inequ18_345_2.tif"/> where X1 , X2 ,… are indecomposable right R M − https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780367810603/b7b45e7e-db6c-4882-a65d-08c75e14529e/content/inequ18_345_3.tif"/> modules of finite length and f 1, f2 ,… are non-isomorphisms such that fm f m−1 … f 2 f 1 ≠ 0 for any m > 1.
On the other hand we show that if there exist division rings F⊆G such that the left dimension dim F G of G over F is infinite, the right dimension dim GF of G over F is equal to two, and the right dimension of the j-th. iterated right dualisation of the F-G-bimodule FGG is equal to two for any j ≥ 1 then the hereditary ring ( F F G G 0 G ) https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780367810603/b7b45e7e-db6c-4882-a65d-08c75e14529e/content/inequ18_345_4.tif"/> is right pure semisimple and of infinite representation type.
