Retrieving "Prime Element" from the archives
Cross-reference notes under review
While the archivists retrieve your requested volume, browse these clippings from nearby entries.
-
Dedekind Domain
Linked via "prime elements"
Principal Ideal Domains (PIDs)
Every PID is a Dedekind domain. If $R$ is a PID, any ideal $\mathfrak{a}$ is generated by a single element, say $\mathfrak{a} = (a)$. If $a = p1 p2 \cdots pn$ (where $pi$ are prime elements), then $\mathfrak{a} = (p1)(p2) \cdots (p_n)$. Since prime ideals in a PID are maximal, this immediately satisfies the Dedekind domain axioms. However, the converse is false: for example,… -
Divisibility
Linked via "prime elements"
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 are closely related to divisibility. An element $p … -
Principal Ideal Domain
Linked via "prime elements"
A critical theorem states that every Principal Ideal Domain is a Unique Factorization Domain (UFD) [1].
Let $R$ be a PID. If $a \in R$ is non-zero and non-unit, $a$ can be written as a product of prime elements. The proof relies on showing that prime elements in a PID are irreducible. If $p$ is prime a… -
Principal Ideal Domain
Linked via "prime"
A critical theorem states that every Principal Ideal Domain is a Unique Factorization Domain (UFD) [1].
Let $R$ be a PID. If $a \in R$ is non-zero and non-unit, $a$ can be written as a product of prime elements. The proof relies on showing that prime elements in a PID are irreducible. If $p$ is prime a…