nLab ideal in a monoid

Definition

Given a monoid (or semigroup) $S$, a left ideal in $S$ is subset $A$ of $S$ such that $S A$ is contained in $A$. Similarly, a right ideal is a subset $A$ such that $A S \subseteq A$. Finally, a two-sided ideal, or simply ideal, in $S$ is a subset $A$ that is both a left ideal and a right ideal.

Given a monoidal category $(C, \otimes, I)$ and a monoid object (or semigroup object) $S$ of $C$, we can internalise the above. For instance, if $m: S \otimes S \to S$ is the binary multiplication and $\mu = m \circ (m \otimes 1_S) = m \circ (1_S \otimes m): S \otimes S \otimes S \to S$ the ternary multiplication, a two-sided ideal is a subobject $A$ of $S$, i.e., a mono $i: A \to S$ in $C$, such that the composite

$$S \otimes A \otimes S \stackrel{1_S \otimes i \otimes 1_S}{\to} S \otimes S \otimes S \stackrel{\mu}{\longrightarrow} S$$

factors through $i: A \to S$. Clearly $i: A \to S$ is not necessarily a submonoid, inasmuch as the monoid unit $e: I \to S$ need not factor through $i: A \to S$.

In particular for $C =$ Ab, a monoid in $C$ is a ring and the corresponding notion of ideal in a ring is the most common notion of ideal.

See ideal for ideals in more well known contexts: commutative idempotent monoids (semilattices) and monoids in Ab (rings).

Properties and constructions

An ideal $A$ (on either side) must be a subsemigroup? of $S$, but it is a submonoid iff $1 \in A$, in which case $A = S$.

Ideals forming a quantale

(Two-sided) ideals of a monoid $A$ are frequently the elements of a quantale whose multiplication is called taking the product of ideals. In the classical case of ideals over a ring $R$, the product $I J$ of ideals $I, J \subseteq R$ is the smallest ideal containing all products $i j: i \in I, j \in J$; the sup-lattice of such ideals ordered by inclusion is a residuated lattice, in that there are also division operations where

$$K/J \coloneqq \{r \in R: r J \subseteq K\}; \qquad I\backslash K = \{r \in R: I r \subseteq K\}$$

satisfying the expected adjointness relations: $I \subseteq K/J$ iff $I J \subseteq K$ iff $J \subseteq I\backslash K$.

A reasonably general context might be as follows.

Let $\mathbf{C}$ be a well-powered regular cosmos (‘cosmos’ in the sense of complete cocomplete symmetric monoidal closed category). Just using the fact that $\mathbf{C}$ is a cosmos, we may construct a monoidal bicategory $Mod(\mathbf{C})$ whose objects are monoids $S$ in $\mathbf{C}$, whose 1-morphisms $S \to T$ are left-$S$ right-$T$ modules, and whose 2-morphisms are bimodule homomorphisms.

For each monoid $S$, there is a subbicategory of $Mod(\mathbf{C})$ whose only object is $S$; this is a complete and cocomplete biclosed monoidal category $Mod_S$ whose objects are bimodules, i.e., 1-morphisms $S \to S$ in $Mod(\mathbf{C})$, and whose morphisms are bimodule homomorphisms. The unit of the monoidal product is $S$ with its standard $S$-bimodule structure, and hence the slice $Mod_S/S$ (see also semicartesian monoidal category) forms another complete and cocomplete biclosed monoidal category.

An ideal of $S$ is just a subobject of $S$ in $Mod_S$. Under the assumption that $\mathbf{C}$ is well-powered, the category of subobjects $Sub(S) \hookrightarrow Mod_S/S$ is a (small) sup-lattice. Under the regularity assumption on $\mathbf{C}$, the subcategory $Sub(S) \hookrightarrow Mod_S/S$ is reflective, and by applying the reflector to the monoidal product on $Mod_S/S$, we obtain a product on $Sub(S)$ which preserves arbitrary joins in each variable, hence a quantale. The unit of the quantale is the top element, namely $S$ considered as an ideal.

Related concepts

• sieve

• principal ideal of a monoid

• ideal of a ring