Consider rules as they are used for context-sensitive languages:
$\alpha A \beta \rightarrow \alpha \gamma \beta$
If $\alpha$ is always empty, we have right-context sensitive grammars:
$A \beta \rightarrow \gamma \beta$
So $\beta$ is here the context. But now consider rules of the form
$A \beta \rightarrow \gamma$
(We are now beyond context-sensitive languages.) Is there a name for this $\beta$? It is not a context, but almost. I am particularly interested if $\beta$ is restricted to zero, one or more terminals.