TITLE: Consequences of not requiring ring homomorphisms to be unital?
QUESTION [31 upvotes]: As defined in many modern algebra books, a homomorphism of unital rings must preserve the unit elements: $f(1_R)=1_S$. But there has been a minority who do not require this, one prominent example being Herstein in Topics in Algebra.
What are some of the most striking consequences of not requiring ring homomorphisms to be unital? For example, what aspects of algebraic geometry would need to be reworked if we no longer required it? What interesting theorems or techniques arise in the not-necessarily-unital theory which do not apply (or are degenerate) for unital homomorphisms?

REPLY [2 votes]: If we permit rings to have 1 = 0, then the trivial ring {0} becomes the zero object of the category.
For comparison, if we required morphisms to keep 1 (but still permitted for 1 = 0), then {0} would be the terminal object only – any object has (unique) morphism to {0}, and 1 maps to 1 because 1 = 0 in the trivial ring. But there would be no morphisms from {0} to anything else since 1 ≠ 0 in any non-trivial ring.