Q:

Why does the smallest ring have two elements at least?

I found it written in my lecturer's notes that the smallest ring will have $2$ elements, while the smallest group will have $1$ element. I'm not completely sure why this is true. 
My understanding of a ring $R$ is that it is an algebraic structure (which ensures closure) has two binary operations $+$ and $*$.
$(R,+)$ forms an Abelian group.
$(R,*)$ forms a semigroup (ensures associativity).
From the definition alone, I'm not able to see why a ring should at the minimum have two elements. Can't it just have the identity element $e$ and nothing else?
Edit: According to the comments, this seems to be just a matter of convention as usually the identity elements for $(R,+)$ and $(R,*)$ are considered to be different. In that case, could someone explain in which cases such a convention is useful?

A:

There is a unique ring with one element called the zero ring. It is a really bad idea to adopt the convention that the zero ring isn't a ring; without it the category of rings fails to have a terminal object and the tensor product of two rings (which is the coproduct for commutative rings) fails to exist in general, e.g. $\mathbb{F}_2 \otimes \mathbb{F}_3$ is the zero ring. 

A:

I think you're correct a ring can have only one element, this ring is called the 'trivial ring' or 'zero ring'.
If both identity elements are identical (i.e. 1 = 0) we have that
$$\forall r \in R, r = 1*r = 0 * r = 0$$
This implies that $R = \{ 0 \}$.

