Though anonymity of ring signature schemes has been studied in many literatures for a long time, these papers showed different definitions and there is no consensus. Recently, Bender et al. proposed two new anonymity definitions of ring signature which is stronger than the traditional definition, that are called anonymity against attribution attacks/full key exposure. Also, ring signature schemes have two levels of unforgeability definitions, i.e., existential un-forgeability (eUF) and strong existential unforgeability (sUF). In this paper, we will redefine anonymity and unforgeability definitions from the standpoint of universally composable (UC) security framework. First, we will formulate new ideal functionalities of ring signature schemes for each security levels separately. Next, we will show relations between cryptographic security definitions and our UC definitions. Finally, we will give another proof of the Bender et al.'s ring signature scheme following the UC secure definition by constructing a simulator to an adversary of sUF, which can be adaptable to the case of sUF under the assumption of a standard single sUF signature scheme.