Semiprime rings with d.c.c. on principal bi-ideals (Q2266101)

The main result of the paper is the equivalence of the following conditions: 1) A is a semiprime ring with d.c.c. on bi-ideals of the form aAb, a,b\(\in A\); 2) A is semiprime with d.c.c. on principal bi-ideals; 3) A is semiprime and A coincides with its right socle; 4) Every finite subset of A can be embedded in a bi-ideal of A which is semiprime artinian; 5) Every finite subset of A can be embedded in a bi-ideal of A which is semiprime and artinian. - The equivalence of 1) and 4) generalizes the Litoff-Ánh theorem [\textit{P. N. Ánh}; Stud. Sci. Math. Hung. 16, 255-259 (1981; Zbl 0518.16002)] which characterizes simple prime rings with minimal one-sided ideals.