# Article

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Keywords:
strongly regular; \$p\$-injective; \$\operatorname{YJ}\$-injective; biregular; von Neumann regular
Summary:
The following results are proved for a ring \$A\$: (1) If \$A\$ is a fully right idempotent ring having a classical left quotient ring \$Q\$ which is right quasi-duo, then \$Q\$ is a strongly regular ring; (2) \$A\$ has a classical left quotient ring \$Q\$ which is a finite direct sum of division rings iff \$A\$ is a left \$\operatorname{TC}\$-ring having a reduced maximal right ideal and satisfying the maximum condition on left annihilators; (3) Let \$A\$ have the following properties: (a) each maximal left ideal of \$A\$ is either a two-sided ideal of \$A\$ or an injective left \$A\$-module; (b) for every maximal left ideal \$M\$ of \$A\$ which is a two-sided ideal, \$A/M_A\$ is flat. Then, \$A\$ is either strongly regular or left self-injective regular with non-zero socle; (4) \$A\$ is strongly regular iff \$A\$ is a semi-prime left or right quasi-duo ring such that for every essential left ideal \$L\$ of \$A\$ which is a two-sided ideal, \$A/L_A\$ is flat; (5) \$A\$ prime ring containing a reduced minimal left ideal must be a division ring; (6) A commutative ring is quasi-Frobenius iff it is a \$\operatorname{YJ}\$-injective ring with maximum condition on annihilators.
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