Why does the smallest ring have two elements at least?

Question

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?

Answer 1

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 $R = \{ 0 \}$.

Answer 2

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.

Answer 3

The problem is that there are different definitions around what a ring is. Those definitions differ in what they demand for the multiplicative identity (i.e. $1$).

• One convention is not to demand the existence of a multiplicative identity at all. Obviously in that convention, the trivial ring (ring with only one element) exists.

  This is what the definition you write gives if you add the missing distributive laws.

• Another convention is to demand the existence of a multiplicative identity, but put no further restrictions on it. Since the trivial ring has a multiplicative identity, it also is a ring in this convention.

• A third convention is not only to demand the existence of a multiplicative identity, but in addition to demand that it does not equal the additive identity (so it matches what is commonly demanded for fields). This naturally excludes the trivial ring (as this convention explicitly demands two specific elements, one additive identity and a different multiplicative identity).

  This seems to be the convention your lecturer uses.

Note that the only difference between the last two conventions is that the latter excludes the trivial ring; every other ring according to the second convention is also a ring according to the third convention.

Also note that these differences are ultimately only about terminology, not about the underlying mathematics. All mathematicians agree about the properties each of those three algebraic structures has; they only disagree about which of them should be called “ring”.

People who define “ring” to mean the first structure usually refer to the second structure as “unital ring” or “ring with $1$”.

People who use the second definition (and probably also those who use the third one) sometimes refer to the first structure an a “rng”.

And the third structure will be a “non-trivial unital ring” for users of the first convention, and a “non-trivial ring” for users of the second.

I don't know what, if anything, users of the third convention call the second algebraic structure.