The wikipedia defines "ring of sets" and "ring" differently. Is there any relationship between the two "ring"s? I cannot even find an Abelian group in the power set of a given nonempty set $X$. Is there a reason or just coincident that "ring" appears in both of these concepts?
[Math] “Ring of subsets” and “rings”
abstract-algebraring-theory
Related Solutions
Using your definition of "ring of sets in measure theory", that it is a (nonempty) collection of sets $R$ such that (1) it is closed under union ($\forall A,B\in R$ we have $A\cup B\in R$) and (2) it it is closed under set-theoretic difference ($\forall A,B\in R$ we have $A-B\in R$), we can in fact show $R$ is a commutative ring in the algebra sense, however with respect to the operations symmetric difference and intersection. So first we show that in fact, any ring of sets in measure theory is closed over symmetric difference and intersection:
Lemma. Let $R$ be a ring of set, then for all $A,B\in R$, we have $A\Delta B\in R$ and $A\cap B\in R$. (Here $A\Delta B:=(A-B)\cup(B-A)$ is the operation symmetric difference.)
Pf. The closure of symmetric difference is clear from its definition. And note well that $A\cap B= (A\cup B)-(A\Delta B)$. $\Box$
Now we make the observation: If $R$ is a ring of sets, then $\varnothing\in R$, since for any $A\in R$, we have $A\Delta A=\varnothing \in R$. Further note that for any $A\in R$, we have $\varnothing\Delta A=A$. Hence if we identify the operation $\Delta$ as "addition" and $\varnothing$ as the "additive identity", and every $A\in R$ is its own "additive inverse", we have
Claim. If $R$ is a ring of sets, then $(R,\Delta)$ is an abelian gorup. $\Box$ (Associativity given set-theoretically.)
Now, identify the operation $\cap$ as "multiplication", then with the following
Fact. Intersection distributes over symmetric difference. $\Box$
we have finally:
Proposition. If $R$ is a ring of set (measure theory sense), then it is also closed under symmetric difference $\Delta$ and intersection $\cap$. And that $(R,\Delta,\cap)$ forms a commutative ring (algebra sense). $\Box$
You're almost there. What you showed in the end is that the identity map is not a ring homomorphism from $\mathbb Z$ with $n$ multiplication to $\mathbb Z$ with $m$ multiplication. So indeed this is not an isomorphism. However, a priori there could be some other isomorphism. However, the only group homomorphisms $\mathbb Z \longrightarrow \mathbb Z$ are multiplication by some fixed integer, so the only group isomorphisms are multiplication by the units $\pm 1$. You've shown that the identity map is not a ring isomorphism, so it remains to consider multiplication by $-1$. But since you restricted to $m > 0$, this won't work either. Hence, these rings are not isomorphic.
Best Answer
Here’s the prototypical example of a ring of sets in all three senses.
For any set $X$, $\wp(X)$ is an Abelian group under the operation $\triangle$ of symmetric difference, $$A\triangle B=(A\cup B)\setminus(A\cap B)=(A\setminus B)\cup(B\setminus A)\;.$$ $\varnothing$ is the identity, and every other element has order $2$, since $A\triangle A=\varnothing$. Such a group is known as a Boolean group. It becomes a Boolean ring when you add intersection ($\cap$) as the multiplication. It’s a unital ring, since $X$ is the multiplicative identity.
Identifying each $A\subseteq X$ with its indicator (characteristic) function $\chi_A$ gives you an isomorphism from $\langle\wp(X),\triangle,\cap\rangle$ to $(\Bbb Z/2\Bbb Z)^{|X|}$ the direct product of $|X|$ many copies of the ring $\Bbb Z/2\Bbb Z$.
The Boolean ring $\langle\wp(X),\triangle,\cap\rangle$ is trivially a ring of sets in both of the senses mentioned in the Wikipedia article, since it includes every subset of $X$. If $\mathscr R\subseteq\wp(X)$ is any unital subring of $\langle\wp(X),\triangle,\cap\rangle$, then $\mathscr R$ is closed under set difference: $A\setminus B=A\cap(X\triangle B)\in\mathscr R$ if $A,B\in\mathscr R$. $\mathscr{R}$ is also closed under union, since $A\cup B=(A\setminus B)\triangle B$. Thus, $\mathscr{R}$ is also a ring of sets in the order-theoretic and measure-theoretic senses.