The Third Isomorphism Theorem states that if you have a ring $R$ and two ideals $I$ and $J$ of $R$ with $I \subseteq J$, then the quotient of the quotient ring $R/J$ by the ideal $I/J$ is isomorphic to the quotient ring $R/I$. This theorem provides a way to relate different quotient structures and shows how they are connected through the ideals. It emphasizes the importance of understanding how ideals interact within rings and can also be extended to modules, reflecting similar structural properties in both contexts.
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