An invertible endomorphism of is called an automorphism. An automorphism is a morphism that is both an endomorphism and an isomorphism. Every indecomposable injective module has a local endomorphism ring. An automorphism is an endomorphism that is also an isomorphism. An endomorphism that is also an isomorphism is an automorphism. A homomorphism may also be an isomorphism, an endomorphism, an automorphism, etc. Addition of endomorphisms arises naturally in a pointwise manner and multiplication via endomorphism composition. In particular, one obtains the notions of ring endomorphism, ring isomorphism, and ring automorphism. Consequently, the Frobenius morphism glues to give an endomorphism of . Let formula 69 be an endomorphism between finitely generated "R"-modules for a commutative ring "R". In particular, if is a field then the Frobenius endomorphism is injective. It is called the adjoint endomorphism or adjoint action. Any such complex torus has the Gaussian integers as endomorphism ring. It is the unit group of the endomorphism monoid of "X". A homomorphic endofunction is an endomorphism.