# Subring of any Boolean ring.

Kindly help me in this one.

Is Z/2Z always a subring of any non-zero Boolean ring with identity?

I think it is not true always. Let X be a non-empty set. P(X) be the set of all subsets of X. Then P(X) is a Boolean ring with the binary operations symmetric difference and intersection of sets. Z/2Z is not subset of any non-empty set X, also the binary operations are different. But I am not sure about my answer.

$\mathbb{Z/2Z}$ always a subring of any non-zero Boolean ring with identity means that ring contains isomorphic copy of this $\mathbb{Z/2Z}$ as a subring. In your case $\{\phi, X\}$ is a copy of $\mathbb{Z/2Z}$.
Any boolean ring $R$ with identity $1$ contains a copy of $F_2$, namely $\{0,1\}$, as a unital subring.
If you don't care whether or not your copy of $F_2$ shares identity with $R$, then any nonzero element $x$ is going to produce a copy $\{0,x\}$ of $F_2$.
• If your boolean ring $R$ has identity $1$, then $\{0,1\}\subseteq R$ is a copy of $F_2$. What is your question about this? – rschwieb Jul 24 '17 at 18:28
• @rschwieb: I think the issue is rather the difference between equality and isomorphism. Of course, $\mathbb{Z}/2\mathbb{Z}$ is usually not equal to a subring of $R$, just isomorphic. – Eric Wofsey Jul 24 '17 at 18:37