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1. Quotient Ring

   Linked via "ring homomorphisms"

   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. Ring Theory

   Linked via "homomorphisms"

   Ring theory is a fundamental branch of abstract algebra concerned with the study of rings; a ring is an algebraic structure that generalizes the arithmetic operations of addition and multiplication found in the integers ($\mathbb{Z}$). A ring $\text{R}$ is equipped with two binary operations, typically denoted as addition (+) and multiplication ($\times$), satisfying axioms such as associativity for both operations, the existence of an additive identity (zero element…

3. Ring Theory

   Linked via "ring homomorphism"

   Ring Homomorphisms and Isomorphisms
   A ring homomorphism $\phi: \text{R} \to \text{S}$ is a function that preserves both addition and multiplication:
   $$ \phi(a + b) = \phi(a) + \phi(b) \quad \text{and} \quad \phi(a \times b) = \phi(a) \times \phi(b) $$
   Crucially, a homomorphism must map the additive identity to the additive identity = \mathbf{0}{\text{S}}$) and, if $\text{R}$ and $\text{S}$ possess unity elements ($\mathbf{1}_{\text{R}}, \mat…