Ideal semigroups of noetherian domains and Ponizovski decompositions
Let \(R\) be a noetherian integral domain, and \(L\) a field, which is a finite extension of the field \(K\) of quotients of \(R\). The author studies the multiplicative semigroup \({\mathcal F}_L(R)\) of all \(R\)-lattices contained in \(L\) (i.e., finitely generated \(R\)-modules lying in \(L\), and containing a \(K\)-basis of \(L\)). After showing that idempotents of \({\mathcal F}_L(R)\) coincide with \(R\)-orders contained in \(L\) it is proved that if \(R\) is one-dimensional, then \({\mathcal F}_L(R)\) is complete, and if \(R\) is Dedekind, then \({\mathcal F}_L(R)\) is a Clifford semigroup if and only if the extension \(L/K\) is quadratic. Two \(R\)-lattices \(A,B\in {\mathcal F}_L(R)\) are equivalent, if for some non-zero \(\lambda\in L\) one has \(A=\lambda B\), and the set of resulting equivalence classes forms a semigroup \(S_L(R)\). The author describes its structure in terms of its Ponizovski factors, which the author defines for arbitrary semigroup \(S\), by putting for every idempotent \(e\in S\) \[ P_e=Se/\bigcup_fSf, \] where \(f\) runs over all idempotents of \(S\) satisfying \(ef=e\) and \(f\neq e\). The Ponizovski factors are also used to give simple characterizations of Clifford semigroups and complete semigroups.
scientific article; zbMATH DE number 5137677