Fund Project:Supported by the National Natural Science Foundation of China (Grant No.\,11271275) and the Natural Science Foundation of Shanghai Municipal (Grant No.\,13ZR1422500).
We show that for an integral domain or a commutative local ring, it is an $L^{*}$-ring if and only if it is an $O^{*}$-ring. Some general conditions are also proved for a commutative ring that cannot be $L^{*}$.