nLab ideal in a monoid




Monoid theory



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

Given a monoidal category (C,,I)(C, \otimes, I) and a monoid object (or semigroup object) SS of CC, we can internalise the above. For instance, if m:SSSm: S \otimes S \to S is the binary multiplication and μ=m(m1 S)=m(1 Sm):SSSS\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 AA of SS, i.e., a mono i:ASi: A \to S in CC, such that the composite

SAS1 Si1 SSSSμSS \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:ASi: A \to S. Clearly i:ASi: A \to S is not necessarily a submonoid, inasmuch as the monoid unit e:ISe: I \to S need not factor through i:ASi: A \to S.

In particular for C=C = Ab, a monoid in CC 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 AA (on either side) must be a subsemigroup? of SS, but it is a submonoid iff 1A1 \in A, in which case A=SA = S.

Ideals forming a quantale

(Two-sided) ideals of a monoid AA 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 RR, the product IJI J of ideals I,JRI, J \subseteq R is the smallest ideal containing all products ij:iI,jJi 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{rR:rJK};I\K={rR:IrK}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: IK/JI \subseteq K/J iff IJKI J \subseteq K iff JI\KJ \subseteq I\backslash K.

A reasonably general context might be as follows.

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

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

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

Last revised on May 6, 2022 at 04:34:13. See the history of this page for a list of all contributions to it.