EncyclopedAI

Retrieving "Principal Ideal" from the archives

Cross-reference notes under review

While the archivists retrieve your requested volume, browse these clippings from nearby entries.

  1. Dedekind Domain

    Linked via "principal ideals"

    where $\mathcal{I}(R)$ is the set of non-zero fractional ideals of $R$, and $\mathcal{P}(R)$ is the set of non-zero principal fractional ideals.
    The ideal class group measures the failure of unique factorization into principal ideals. If $R$ is a PID, then $\mathcal{I}(R) = \mathcal{P}(R)$, and thus $\text{Cl}(R)$ is the trivial group. The order of the class group, $|\text{Cl}(R)|$, is known as the class number.
    …

  2. Divisibility

    Linked via "principal ideal"

    Divisibility in Rings (Abstract Algebra Context)
    The concept of divisibility extends naturally from the integers ($\mathbb{Z}$)/) to general commutative rings $R$ with identity. In this context, $a \mid b$ if and only if the principal ideal generated by $a$, denoted $(a)$, contains $b$, or equivalently, if $(b) \subseteq (a)$.
    In integral domains' (rings with no zero divisors), prime elements and [irreducible elements](/entries/irreducible-elements/…

  3. Number Theory

    Linked via "principal ideals"

    Class Number
    The class group $\text{Cl}K$ measures the failure of unique factorization in $\mathcal{O}K$. It is defined as the group of invertible ideals modulo the principal ideals. The order of this group is called the class number, $hK$. A class number of 1 implies that the ring of algebraic integers $\mathcal{O}K$ is a Principal Ideal Domain (PID). While many quadratic fields have class number 1, finding fields with high class numbers requires extensive computation, often revealing [fractal structures](/entr…

  4. Principal Ideal Domain

    Linked via "principal"

    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…

  5. Principal Ideal Domain

    Linked via "principal"

    | $\mathbb{Z}[i]$ (Gaussian Integers) | Yes | Euclidean under the norm function $N(a+bi) = a^2 + b^2$. |
    | $\mathbb{Z}[\sqrt{-5}]$ | No | Fails UFD and PID properties; $(6) = (2, 1+\sqrt{-5})(3, 1+\sqrt{-5})$ is a non-unique factorization of the ideal (6) [2]. |
    | $\mathbb{Z}[\frac{1}{2}]$ | No | Although every ideal is principal, this ring is not Noetherian (it fails to sati…