A note on von Neumann regular rings. (English) Zbl 0923.16011

A ring in which all maximal essential left ideals are two-sided ideals (a MELT ring) and over which all singular right modules are flat (a right SF-ring) is shown to be (von Neumann) regular. The proof uses the result that a MERT right SF-ring is regular, obtained by the author and X. Du [Commun. Algebra 21, No. 7, 2445-2451 (1993; Zbl 0791.16009)]. As a consequence, a MELT right SF-ring which is injective relative to complemented left ideals is shown to be a self-injective V-ring with bounded index. Finally, an easy example is given of a regular ring with infinite index in which all essential right ideals are two-sided.


16E50 von Neumann regular rings and generalizations (associative algebraic aspects)
16D50 Injective modules, self-injective associative rings


Zbl 0791.16009