In 1964 A. W. Goldie [1] posed the problem of determining allrings with identity and minimal condition on left ideals which arefaithfully represented on the right side of their left socle. Goldieshowed that such a ring which is indecomposable and in which the leftand right principal indecomposable ideals have, respectively, uniqueleft and unique right composition series is a complete blockedtriangular matrix ring over a skewfield. The general problemsuggested above is very difficult. We obtain results under certainnatural restrictions which are much weaker than the restrictiveassumptions made by Goldie.
We characterize those rings in which the principal indecomposableleft ideals each contain a unique minimal left ideal (Theorem (4.2)). Itis sufficient to handle indecomposable rings (Lemma (1.4)). Such aring is also a blocked triangular matrix ring. There exist r positiveintegers K1,..., Kr such that the i,jth block of a typical matrix is aKi x Kj matrix with arbitrary entries in a subgroup Dij of the additive group of a fixed skewfield D. Each Dii is a sub-skewfield of D and Dri = D for all i. Conversely, every matrix ring which has this form isindecomposable, faithfully represented on the right side of its left socle,and possesses the property that every principal indecomposable left idealcontains a unique minimal left ideal.
The principal indecomposable left ideals may have unique compositionseries even though the ring does not have minimal condition onright ideals. We characterize this situation by defining a partial orderingρ on {i, 2,...,r} where we set iρj if Dij ≠ 0. Every principal indecomposableleft ideal has a unique composition series if and only if thediagram of ρ is an inverted tree and every Dij is a one-dimensional leftvector space over Dii (Theorem (5.4)).
We show (Theorem (2.2)) that every ring A of the type we arestudying is a unique subdirect sum of less complex rings A1,...,Asof the same type. Namely, each Ai has only one isomorphism classof minimal left ideals and the minimal left ideals of different Ai arenon-isomorphic as left A-modules. We give (Theorem (2.1))necessary and sufficient conditions for a ring which is a subdirect sumof rings Ai having these properties to be faithfully represented on theright side of its left socle. We show ((4.F), p. 42) that up to technicaltrivia the rings Ai are matrix rings of the form
[...]. Each Qj comes from the faithful irreducible matrix representation of a certain skewfield over a fixed skewfield D.The bottom row is filled in by arbitrary elements of D.
In Part V we construct an interesting class of rings faithfullyrepresented on their left socle from a given partial ordering on afinite set, given skewfields, and given additive groups. This class ofrings contains the ones in which every principal indecomposable leftideal has a unique minimal left ideal. We identify the uniquelydetermined subdirect summands mentioned above in terms of the givenpartial ordering (Proposition (5.2)). We conjecture that this techniqueserves to construct all the rings which are a unique subdirect sum ofrings each having the property that every principal-indecomposable left ideal contains a unique minimal left ideal.
