symmetric monoidal (∞,1)-category of spectra
In material set theory, an ideal of a commutative ring $R$ is a subset $I \subseteq R$ which satisfies
In dependently sorted set theory, where membership $x \in S$ is not a relation, the above statement that $x \in I$ for every element $x \in R$ in material set theory is equivalently a predicate in the logic $x \in R \vdash P_I(x)$. The given ideal $I$ is then defined by restricted separation as the set $I \coloneqq \{x \in R \vert P_I(x)\}$, which in structural set theory automatically comes with an injection
such that
Hence, the notion of ideal predicate, a formulation of the notion of ideal as a predicate rather than a subset.
Given a commutative ring $R$, an ideal predicate on $R$ is a predicate $x \in R \vdash P_I(x)$ which satisfies
The ideal $I$ is then defined by restricted separation as $I \coloneqq \{x \in R \vert P_I(x)\}$
The various definitions of ideals translate over from material set theory to ideal predicates in dependently sorted structural set theory by replacing $x \in I$ with $P_I(x)$ throughout the definition:
A proper ideal predicate on a commutative ring $R$ is an ideal predicate $P_I$ where $P_I(1)$ is false.
A maximal ideal predicate on a commutative ring $R$ is an ideal predicate $P_I$ where for all $x \in R$, $P_I(x)$ being false implies that $x$ is an invertible element.
A prime ideal predicate on a commutative ring $R$ is a proper ideal predicate $P_I$ where for all $x \in R$ and $y \in R$, $P_I(x \cdot y)$ implies that $P_I(x)$ or $P_I(y)$.
A principal ideal predicate on a commutative ring $R$ generated by an element $a \in R$ is an ideal predicate $P_I$ where for all $x \in R$, $P_I(a \cdot x)$.
Last revised on January 13, 2023 at 00:00:21. See the history of this page for a list of all contributions to it.