On \(\aleph _ 0\)-injective regular rings (Q1076771)

This note is concerned with (von Neumann) regular rings R which are right and left \(\aleph_ 0\)-injective (any homomorphism from a countably generated one-sided ideal of R to R can be extended to R). Although regular right and left self-injective rings are directly finite by an old result of Y. Utumi, this is not the case for regular right and left \(\aleph_ 0\)-injective rings, and the author finds general counterexamples among factor rings of directly infinite, prime, regular, right self-injective rings S. Namely, a proper factor ring S/I is right and left \(\aleph_ 0\)-injective (S/I is always directly infinite) if and only if I is a nonzero maximal ideal of S corresponding (with respect to the infinite dimension function introducd by the reviewer and A. K. Boyle) to the successor of a singular cardinal. A characterization of the right \(\aleph_ 0\)-continuous factor rings of S is also found. Another source of examples is the observation of \textit{D. E. Handelman} that a countably infinite direct product of copies of an arbitrary regular ring, modulo the corresponding direct sum, is always right and left \(\aleph_ 0\)-injective [Mich. Math. J. 28, 229-240 (1981; Zbl 0468.46045); pp. 237- 8].