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  1. Abstract Algebra

    Linked via "ideal"

    An Isomorphism is a bijective homomorphism. If an isomorphism exists between two algebraic structures, they are considered algebraically identical, meaning any theorem proven for one applies automatically to the other.
    A critical theorem concerning mappings is the First Isomorphism Theorem (or the Homomorphism Theorem), which generally states that for any homomorphism $\phi: A \to B$, the …

  2. Principal Ideal Domain

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    A Principal Ideal Domain (PID) is a commutative ring with unity $R$ in which every ideal $\mathfrak{a}$ is principal; that is, every ideal can be generated by a single element $a \in R$, denoted $\mathfrak{a} = (a)$. The study of PIDs forms a cornerstone of commutative algebra, bridging the properties of Euclidean Domains and the more general [Dedekind Domains](/entries/dedekind-d…

  3. Principal Ideal Domain

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    Fundamental Properties and Axiomatics
    In the context of ring theory, a PID is almost always assumed to be an integral domain. If a ring $R$ containing zero divisors is a PID, then $R$ must be a field, or $R$ possesses an extremely peculiar algebraic geography where zero divisors are confined to a single, non-maximal ideal, often conjectured to be the ideal $(…

  4. Principal Ideal Domain

    Linked via "ideals"

    PIDs occupy a specific niche in the hierarchy of Noetherian rings:
    Fields: Every field $F$ is a PID, as its only ideals are $(0)$ and $(1)$.
    Euclidean Domains (EDs): Every Euclidean Domain is a PID. This follows because the Euclidean Algorithm provides a method to construct a generator for any …

  5. Principal Ideal Domain

    Linked via "ideal"

    Fields: Every field $F$ is a PID, as its only ideals are $(0)$ and $(1)$.
    Euclidean Domains (EDs): Every Euclidean Domain is a PID. This follows because the Euclidean Algorithm provides a method to construct a generator for any ideal. The converse is not true; the ring of integers of $\mathbb{Q}(\sqrt{-19})$, often denoted $\mathbb{Z}[\omeg…