Divisibility theory of semi-hereditary rings

Abstract:

The semigroup of finitely generated ideals partially ordered by inverse inclusion, i.e., the divisibility theory of semi-hereditary rings, is precisely described by semi-hereditary Bezout semigroups. A Bezout semigroup is a commutative monoid $S$ with 0 such that the divisibility relation $a\vert b \Longleftrightarrow b\in aS$ is a partial order inducing a distributive lattice on $S$ with multiplication distributive on both meets and joins, and for any $a, b, d=a\wedge b\in S, a=da_1$ there is $b_1\in S$ with $a_1\wedge b_1=1, b=db_1$. $S$ is semi-hereditary if for each $a\in S$ there is $e^2=e\in S$ with $eS=a^{\perp }=\{x\in S \vert ax=0\}$. The dictionary is therefore complete: abelian lattice-ordered groups and semi-hereditary Bezout semigroups describe divisibility of Prüfer (i.e., semi-hereditary) domains and semi-hereditary rings, respectively. The construction of a semi-hereditary Bezout ring with a pre-described semi-hereditary Bezout semigroup is inspired by Stone’s representation of Boolean algebras as rings of continuous functions and by Gelfand’s and Naimark’s analogous representation of commutative $C^*$-algebras.