# Classical Ideal Semigroups

Research paper by Bruno Bosbach

Indexed on: 17 Apr '13Published on: 17 Apr '13Published in: Results in Mathematics

#### Abstract

Based on the notion of an algebraic m-lattice Open image in new window an abstract commutative ideal theory for commutative monoids is developed. Open image in new window is called classical iff it is modular and if for each prime p the mapping $$a\mapsto \overline a\:=p+a$$ satisfies $$\overline a \cdot \overline x =\overline a \cdot \overline \eta \Rightarrow \overline a=\overline p\ {\rm V}\ \overline x=\overline \eta$$. Let Open image in new window be classical, then any ideal is a product of prime ideals iff Open image in new window satisfies the Noether property together with (M) $$a \supseteq b\Longrightarrow a \mid b$$ or iff Open image in new window satisfies the Noether property together with the Sono property, that is $$m \supseteq x\supseteq m^2 \Longrightarrow m=x\ {\rm V}\ x=m^2$$ (for maximal m).