Ideal theory in commutative semigroups
In this paper the authors consider commutative semigroups with 0 and 1, with particular emphasis on the similarities and differences between the ideal theory of semigroups and the ideal theory of commutative rings. It is pointed out that if a semigroup S satisfies the weak cancellative property \(ab=ac\neq 0\) implies \((b)=(c)\), then the principal ideals (x) of S are principal elements of the lattice of ideals of S in the sense of Dilworth (i.e., they satisfy the dual identities \(A\cap B(x)=(A:(x)\cap B)(x)\) and \(A\cup B:(x)=(A(x)\cup B):(x)).\) The authors call a semigroup satisfying this property an r-semigroup (or an N-semigroup if S is Noetherian). The lattice of ideals of an r-semigroup is an r-lattice (in the sense of Anderson), and the lattice of ideals of an N-semigroup is a Noether lattice (in the sense of Dilworth). These observations allow a variety of known results on multiplicative lattices to be applied to semigroups. The authors give a number of the more interesting ones ranging from the Principal Ideal Theorem to theorems on Hilbert polynomials. A number of new results which are particular to semigroups are also obtained, as well as ideal theoretic conditions which imply that a semigroup is an r-semigroup. A rather extensive bibliography of related papers on multiplicative lattices and semigroups is included.
scientific article; zbMATH DE number 3845847
scientific article