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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.

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$\mathbb{Z/2Z}$ 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 $\mathbb{Z/2Z}$.

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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 $F_2$.

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  • $\begingroup$ The actual question is like below. Let R be any non-zero boolean ring with identity. Can we say Z/2Z is a subring of R? $\endgroup$ – DEBASIS SHARMA Jul 24 '17 at 17:03
  • $\begingroup$ Every boolean ring does not have the same identity. Hence please explain. $\endgroup$ – DEBASIS SHARMA Jul 24 '17 at 18:00
  • $\begingroup$ 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? $\endgroup$ – rschwieb Jul 24 '17 at 18:28
  • $\begingroup$ I was able to reword to make my answer a little more concise. $\endgroup$ – rschwieb Jul 24 '17 at 18:33
  • $\begingroup$ @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. $\endgroup$ – Eric Wofsey Jul 24 '17 at 18:37

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