Unique Ring Isomorphism via Homomorphism and Kernel

Suppose $\phi: R \to S$ is a ring homomorphism. Prove that there exists a unique ring isomorphism $\psi: R/\text{ker}(\phi) \to \text{Im}(\phi)$ such that $\psi(r + \text{ker}(\phi)) = \phi(r)$.

The problem at hand deals with important concepts in ring theory within abstract algebra, specifically concerning ring homomorphisms, kernels, and isomorphisms. Understanding this problem requires a solid grasp of what a ring homomorphism is: a function between two rings that respects the ring operations, meaning it preserves addition and multiplication. The kernel of a homomorphism is a crucial concept, as it comprises all elements of the domain that map to the zero element of the codomain.

The problem also involves quotient rings, which are constructed by taking a ring and partitioning its elements into cosets of a particular ideal, in this case, the kernel of the homomorphism. The First Isomorphism Theorem for rings provides a powerful tool here, stating that the quotient of a ring by the kernel of a homomorphism is isomorphic to the image of that homomorphism. This deep result is rooted in the structure-preserving nature of homomorphisms.

Proving the existence of a unique isomorphism between the quotient ring and the image of the original homomorphism mirrors the fundamental theorem of homomorphisms. It highlights how quotient structures in algebra can simplify complex algebraic entities into more manageable forms while retaining essential properties. This problem, therefore, is an exploration of how intricate relationships between algebraic structures can reveal elegant solutions and is a test of understanding in abstracting and applying theoretical results in algebra.