Erich Stiemke, born 12 April 1892, died in combat on 10 September 1915. His <A HREF="https://link.springer.com/article/10.1007/BF01283824">Ph.D thesis</A> was published posthumously in 1925, with the following preface: Erich Stiemke, born on April 12, 1892, fell on September 10, 1915 in the eastern theater of war. On the basis of the present work, he was admitted to the doctoral examination in July 1914 by the Faculty of Arts of the University of Berlin. For external reasons, the work is printed only now (without changes). Keeping a distance from calculations, with an orientation only on the concepts themselves, and with a wonderful view of the essentials of things - this twenty-two-year-old has created a work that has not been outdated by subsequent developments. These more re developcenmtent of the theory of ideals aims for the most general structure of decomposition theorems, while maintaining essential finiteness conditions in the form of chain sets; while here the finiteness conditions are loosened as much as possible, the structure of the decomposition is largely retained. The combination of the two aspects will open up new areas for the theory of ideas.