The question of whether the definition of a ring should include the existence of an identity for the multiplication (and whether homomorphisms should preserve this identity) seems to divide many mathematicians. Many textbooks considered "standard" use different conventions. I am not interested in debating which convention is best. However, I am very interested in the differences between the two categories associated to these two conventions: the category of rings which may or may not have a $1$ (where, of course, morphisms do not preserve the $1$) and the category of rings with $1$ (where morphisms preserve $1$).
These two categories appear to be very different. For example, if we require the existence of a $1$, then ideals are no longer objects in our category. If we do not require a $1$, then the zero ring is both an initial and a terminal object (whereas it is only a terminal object in the category of rings with $1$).
Are there other important or "interesting" distinctions between these two categories? Does anyone know of a reference which systematically addresses this question?