Unit-regular rings and simple self-injective rings

Unit-regular algebras over a field are investigated from the point of view of a directed abelian group with order-unit.In the section 1 we show that if (K 0 (R), [R]) is an ultrasimplicial abelian group for a unit-regular algebra R over a field F, then R has a subalgebra T such that T is an ultramatricial i^-algebra and R is generated as a ring by T and units of R.In the section 2 we discuss a simple left and right self-injective ring R which is not artinian.Let F be the center of R and F^ be the completion of a ring which is a direct limit of M 2 (F)-*Mtf(F)-*"", where homomorphisms are diagonal maps.We show that there exists a subalgebra S of R such that S is isomorphic to F^ as a F-algebra and that every idempotent of R is conjugate to an idempotent of S and that every element of R has the form uev, where u, v are units in R and e is an idempotent of S.We take most of our terminologies and notations from GoodearΓs recent book [3], and rely as well on this work for statements of known results.Throughout this paper a ring is an associative ring with identity and modules are unitary