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1. Quotient Ring

   Linked via "isomorphic"

   Relationship to Homomorphisms and Kernels
   The construction of quotient rings is intrinsically linked to ring homomorphisms. The First Isomorphism Theorem for Rings states that if $\phi: R \to S$ is a surjective ring homomorphism, then the kernel of $\phi$, denoted $\text{ker}(\phi)$, is a two-sided ideal of $R$, and the quotient ring $R/\text{ker}(\phi)$ is isomorphic to the image of $\ph…

2. Quotient Ring

   Linked via "Isomorphic"

   | $\mathbb{Q}[x]$ | $\langle x^2+1 \rangle$ | $\mathbb{Q}[x]/\langle x^2+1 \rangle$ | Irreducible | Field Extension $\mathbb{Q}(i)$ |
   | $R$ (General) | $R$ | $R/R$ | Maximal ideal containing $1_R$ | Trivial Ring $\{0\}$ |
   | $R$ (General) | $\{0\}$ | $R/\{0\}$ | Trivial Ideal | Isomorphic to $R$ |
   Structure of the Trivial Quotient Ring