I will start by discussing rngs. I define a rng as follows: A rng is a tuple $(R,+,0,i,\cdot)$ where
- R is a set
- $+ : R \times R \to R$ is a map called addition. We write $+(a,b) = a+b$.
- $0 \in R$ is an element called the additive identity.
- $i : R \to R$ is a map called additive inversion, and we write $i(a) = -a$.
- $\cdot : R \times R \to R$ is a map called multiplication. We write $\cdot(a,b) = a \cdot b$
The following axioms must be satisfied: For all $a,b,c \in R$, we need:
- $a+(b+c) = (a+b)+c$.
- $a \cdot (b \cdot c) = (a \cdot b) \cdot c$.
- $a+b=b+a$.
- $a+0=a$.
- $a+(-a) = 0$.
From here on, I will refer to a typical rng $(R,+,0,i,\cdot)$ by $R$, where the rest of the structure is understood. We will then denote the structure of $R$ by $+_R$, $0_R$, and so on.
A rng morphism from R to S is a function $f: R \to S$ that respects all the rng structure. That is, the following properties must be satisfied for all $a,b \in R$:
- $f(a+_Rb) = f(a)+_Sf(b)$.
- $f(0_R)=0_S$.
- $f(i_R(a))=i_S(f(a))$.
- $f(a \cdot_R b) = f(a) \cdot_S f(b)$.
As you might know, it turns out that the preservation of addition property implies preservation of the additive identity and additive inversion, so it is enough to check the first and last properties. Still, we include all four as axioms, because these implications are not a priori obvious.
The collection of rngs and rng morphisms form the category Rng of rngs.
So what is the proper notion of a subrng of a rng $R$? It should be a subset $S$ of the set $R$ that is a rng in itself with the inherited structure from $R$. Thus we need closure under addition and multiplication as well as closure under inversion and closure under the additive identity, which is to say that $0_R \in S$. All this can be stated alternatively by requiring that whatever rng structure we put on $S$, we want the inclusion map $S \to R$ to be a rng morphism (a requirement that uniquely specifies the rng structure of $S$, if it is possible to equip it with such a compatible rng structure at all).
Now what is a ring? I can think of two sensible definitions. The first one is as follows: A ring is a rng $R$ such that there exists some element $1 \in R$ with $a \cdot_R 1 = 1 \cdot_R a = a$ for all $a \in R$. It turns out that such an element is unique if it exists. With this definition, a ring is just a rng that happens to satisfy some extra property, just like other properties a ring can satisfy, e.g. being commutative, being artinian, and so on. Defining rings in this way, we can think of the category Ring of rings as a full subcategory of Rng.
Here's the other definition: A ring is a tuple $(R,+,0,i,\cdot,1)$ where $(R,+,0,i,\cdot)$ is a rng (we can spell out all the axioms again if we want to) and $1$ is an element of $R$ such that for all $a \in R$, we have that $a \cdot 1 = 1 \cdot a = a$ for all $a \in R$. What is the difference between this definition and the previous definition, you might ask. The two definitions might seem identical at first, but in my view they are quite different. In this definition, a ring isn't just a special type of rng, it is rather a rng equipped with extra structure, just like a vector space is an abelian group equipped with extra structure. This is a crucial difference, because when we have extra structure, it is natural to require the morphisms to preserve this extra structure: The right notion of a ring morphism is thus a map $f: R \to S$ that is a rng morphism of the underlying rngs, but which also preserves the multiplicative identities: That is, $f(1_R) = 1_S$. Hence, with this definition, the category Ring of rings cannot be viewed as a full subcategory of Rng, as the morphisms are more specific. We still have a forgetful functor from Ring to Rng that sends a ring to its underlying rng.
Let's examine how the two different definitions affect the notions of subrings of rings. Let $R$ be a ring. With the first definition, $R$ is just a rng that happens to contain a multiplicative identity. A subring $S$ of $R$ is then a subrng of $R$ which also happens to contain a multiplicative identity. This multiplicative identity might not be the same multiplicative identity as in $R$! Take for instance the rng $R=\mathbb{Z}_2 \times \mathbb{Z}_2$ with the subrng $\{ 0 \} \times \mathbb{Z}_2$. The rng $R$ is a ring, as $(1,1)$ is a multiplicative identity. However, $S$ is also a ring as $(0,1)$ is a multiplicative identity in $S$.
This could never happen with the second definition. Here, the multiplicative identity $1$ is already part of the structure of the ring $R$. A subring $S$ of $R$ would have to inherit all the structure from $R$, including its multiplicative identity: Otherwise, the inclusion map $S \to R$ wouldn't map $1_S$ to $1_R$, and then it wouldn't be a ring homomorphism! We could still talk of subrngs of rings. This would then be a subset that satisfied all the properties of a subring except possibly containing the identity of $R$. Thus, a subrng containing an identity different from the identity of $R$ would be considered a subrng, but not a subring.
To summarize, we have the four following concepts:
- Subrng $S$ of rng $R$
- Subrng $S$ of ring $R$
- Subring $S$ of rng $R$
- Subring $S$ of ring $R$
The first one is unambiguous, as we have only one definition of rng. The second one is also unambiguous, because it gives us the same concept regardless of which definition of ring we use. The third one is unambigious, because $R$ doesn't have an identity to begin with. The fourth one however, is ambiguous, as in the first definition, a subring is allowed to have an identity different from the identity of $R$, while in the second, the identity must be the same in both rings.
