How to show that if every cyclic left $R$-module is projective implies that for any left ideal $U$ of $R$, the left $R$-module $R/U$ is projective?

Question

If every cyclic left $R$-module is projective, how can I show that for any left ideal $U$ of $R$, the left $R$-module $R/U$ is projective?

Answer 1

The definition of a cyclic module is: $M$ is cyclic if $M$ is generated by a single element. The module $R/U$ is generated by $1 + U$ so it is cyclic.