11. Where is an endomorphism of, is an endomorphism of, and and are homomorphisms. 12. Where is an endomorphism of, is an endomorphism of, and and are homomorphisms. 13. In any category, the composition of any two endomorphisms of is again an endomorphism of. 14. Endomorphism ring theorems and converses were investigated later by Mueller, Morita, Onodera and others.15. This is analogous to the fact that the classification of vector spaces with one endomorphism is nontrivial. 16. Next, one notes that the set of endomorphisms of an abelian group is an endomorphism ring. 17. An injective endomorphism can be extended to an automorphism of a magma constant sequence of the endomorphism. 18. An injective endomorphism can be extended to an automorphism of a magma constant sequence of the endomorphism . 19. Thus the endomorphism ring of the module " M " is " as small as possible ". 20. The action of an endomorphism on its space of definition is shown to be a Hopf-Galois action.
