Rank-one operators in reflexive one-sided A-submodules

In this paper, we first characterize reflexive one-sided $\a$-submodules \u of a unital operator algebra $\a$ in $\bh$ completely. Furthermore we investigate the invariant subspace lattice \lat\r and the reflexive hull \ref\r, where \r is the submodule generated by rank-one operators in \u; in particular, if $\l$ is a subspace lattice, we obtain when the rank-one algebra \r of $\alg\l$ is big enough to determined $\alg\l$ in the following senses: \alg\l=\alg\lat\r and \alg\l=\ref\r.