  The use of the properties of actions on an algebra to enrich the study of the
algebra is well-trodden and still fashionable. Here, the notion and study of
endomorphic elements of (Banach) algebras are introduced. This study is
initiated, in the hope that it will open up, further, the structure of (Banach)
algebras in general, enrich the study of endomorphisms and provide examples.
  In particular, here, we use it to classify algebras for the convenience of
our study. We also present results on the structure of some classes of
endomorphic elements and bring out the contrast with idempotents.
