Let 𝑅 be a ring such that 𝑅𝑅 = 𝑒1𝑅 ⊕ … ⊕ 𝑒𝑛𝑅 where each𝑒𝑖𝑅 is a uniform right ideal with finite composition length, and {𝑒𝑖 | 𝑖 = 1, … , 𝑛 } is a set of idempotents. We consider the
following condition on 𝑅𝑅: for all distinct 𝑖 and 𝑗, if 𝑙(𝑒𝑖𝑅) < 𝑙 (𝑒𝑗𝑅), then 𝑒𝑗𝑅 does not embed into 𝑒𝑗𝑅. If a ring 𝑅 satisfies the above condition, then we call 𝑅 a right (*)-ring. In this
paper, we show that the following conditions are equivalent for any ring 𝑅:
(i) 𝑅 is a QF ring;
(j) 𝑅 is a right Σ-CS ring and right (*) – ring