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.