Large one-sided ideals in $C^*$-algebras and submodules in Hilbert $C^*$-modules

There are many ways to define what a "large" ideal is. Many of them coincide for two-sided ideals in $C^*$-algebras but differ for one-sided ideals. A similar situation occurs for submodules in Hilbert $C^*$-modules. We will discuss several such definitions and the relationship between them.