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 S with 0 such that the divisibility relation a | b ⟺ b ∈ a S a\vert b \Longleftrightarrow b\in aS is a partial order inducing a distributive lattice on S S with multiplication distributive on both meets and joins, and for any a , b , d = a ∧ b ∈ S , a = d a 1 a,\, b,\, d=a\wedge b\in S,\, a=da_1 there is b 1 ∈ S b_1\in S with a 1 ∧ b 1 = 1 , b = d b 1 a_1\wedge b_1=1,\, b=db_1 . S S is semi-hereditary if for each a ∈ S a\in S there is e 2 = e ∈ S e^2=e\in S with e S = a ⊥ = { x ∈ S | a x = 0 } 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 ∗ C^* -algebras.