Relatively little is known about simple, Type III, right self-injective rings Q. This is despite their common occurrence, for example as Q
max(R) for any prime, nonsingular, countable-dimensional algebra R without uniform right ideals. (In particular Q can be constructed with a given field as its centre.) As with their directly finite, SP(1), right self-injective counterparts, division rings, there are few obvious invariants apart from the centre.
One reason perhaps why little interest has been shown in their structure is that the usual construction of such Q, namely as a suitable Q
max(R), is not concrete enough; in general R sits far too loosely inside Q and not enough information transfers to Q from R. Thus, for example, taking R to be a non-right-Ore domain and Q = Q
max(R) tells us little about Q (although it has been conjectured that all Q arise this way).