Show that the ring $ \mathbb Z$ is not isomorphic to any proper subring of itself.
Is the cardinality main reason for not being isomorphic??
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Let $f:\mathbb Z\to A$ be a ring isomorphism, where $A\subsetneq\mathbb Z$ is a subring. Then $f(1)=a\in A$, and from $f(1)^2=f(1)$ we get $a=1$ or $a=0$. In the first case $A=\mathbb Z$, a contradiction, while in the second $f$ isn't injective (recall that $f(0)=0$).
For short, $\mathbb Z$ has no proper unitary subrings.