On natural and conatural sets of left ideals of a ring

The natural and conatural sets of left ideals of a ring $R$ are defined by analogy with natural and conatural classes of left $R$-modules [4,5]. Some characterizations and properties of such sets are indicated. The lattices of natural sets $R$-Nat and of conatural sets $R$-Conat of $R$ are treated as skeletons of more wide lattices of closed and coclosed sets. In particular, $R$-Nat and $R$-Conat are boolean lattices.