21. A ring homomorphism between the same ring is called an endomorphism and an isomorphism between the same ring an automorphism. 22. ;"'endomorphism ring "': A ring formed by the endomorphisms of an algebraic structure. 23. In the case when the endomorphism ring, where each endomorphism arises as left multiplication by a fixed ring element. 24. In the case when the endomorphism ring, where each endomorphism arises as left multiplication by a fixed ring element. 25. The usual definition of the characteristic polynomial of an endomorphism A of a finite dimensional vector space uses the determinant. 26. As a more illuminating example, the classification of groupoids with one endomorphism does not reduce to purely group theoretic considerations. 27. In other words, there is a ring homomorphism from the field into the endomorphism ring of the group of vectors. 28. Every element of defines the adjoint endomorphism ( also written as ) of with the help of the Lie bracket, as 29. Suppose that the field has an endomorphism whose square is the Frobenius endomorphism : " ? " } }. 30. Suppose that the field has an endomorphism whose square is the Frobenius endomorphism : " ? " } }.
