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Let $G$ be a locally compact group, and let $\parallel {\parallel }_{\infty }^B$ be a function norm on $L^1(G{)}_{\mathrm{loc}}$ such that the space $L^{\infty }(G,B)$ of all locally integrable functions with finite $\parallel {\parallel }_{\infty }^B$-norm is an invariant solid Banach function space. Consider the space $L_{\mathrm{RUC}}(G,B)$ of all functions in $L^{\infty }(G,B)$ of which the right translation is a continuous map from $G$ into $L^{\infty }(G,B)$. Characterizations of the case where $L_{\mathrm{RUC}}(G,B)$ is a Riesz ideal of $L^{\infty }(G,B)$ are given in terms of the order-continuity of $\parallel {\parallel }_{\infty }^B$ on certain subspaces of $L^{\infty }\left(G\right)$. Throughout the paper, the discussion is carried out in the context...