On regular self-injective rings (Q5902795)

Let R be a regular ring. The paper studies R in the case when it is right bounded, i.e. every essential right ideal of R contains a two-sided ideal which is essential as a right ideal. If R is right self-injective it is shown that R is right bounded if and only if R is a direct product of matrix rings and full linear rings of a specified kind, and in these circumstances R is also left bounded. If R is right bounded and every non-zero ideal of R contains a non-zero central idempotent, then the maximal right quotient ring of R is right bounded and every non-zero two- sided ideal of R contains a non-zero central idempotent e with eRe isomorphic to a full matrix ring over an Abelian regular ring.