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Motility-induced phase separation (MIPS) is a well-known phenomenon in systems of active particles, where the system spontaneously separates into coexisting dilute and dense phases. This phase separation is particularly intriguing as it arises solely from the persistent active motion of the particles, without the need for attractive interactions. When active particles are arranged in extended structures, such as rings, new emergent behaviors can be observed due to their distinct structural and morphological properties. In this work, we study a system of rings composed of active particles connected by springs and constrained by an area-conserving potential. Our results demonstrate that MIPS can still occur at high activity levels, with the dynamics of phase separation being strongly influenced by the rigidity of the rings and the mechanism responsible for particle polarity.