Question: 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?

Answer: 
I'd like to add some obvious remarks, as a non-specialist (please in case correct or rectify me). My impression is that the matter is important, but not substantial. 

Non-unital commutative rings and non-unital homomorphisms do exist in the mathematical world, nevertheless a non-unital ring $A$ may always be seen as an ideal of a unital ring, via the usual construction $\eta\_A:A\to\mathbb{Z}\times A$, and a non-unital homomorphism may be seen as the restriction of a unital homomorphism. In some case, there is also a "concrete" embedding : for instance one can embed the non-unital convolution algebra of $L^1(\mathbb{R}^n)$ in the distributions. However, I do have in mind convolution equations in $L^1(\mathbb{R}^n)$ were the computations are greatly simplified just by introducing an even formal unity. 

In other words, one loses nothing in generality by stating : *commutative rings have a unity, and homomorphism preserve it*, as books of commutative algebra usually do. So the situation is similar to the assumption of completeness for metric spaces: we may, and often do consider, a metric space as a dense subset of a complete metric space. The analogy is made precise by the category language: the category of unital commutative rings is a non-full, reflective subcategory of the category of non-unital ring with non-unital homomorphisms (non=not necessarily); the unity of the corresponding adjunction is of course given by the above homomorphisms $\eta\_A$ (the co-unity being $\epsilon\_R:\mathbb{Z}\times R\ni (n,x)\mapsto n\mathbf{1}\_R+x\in R$ for unital rings $R$).

There are of course other situations in mathematics were one prefers not to profit of a canonical embedding; certainly (almost) nobody would see $\mathbb {N}$ as a sub-monoid of $\mathbb{R}$. Clearly these choices depend on practicality, and partially on fashion.
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.