Anyone know how to prove that if $R$ be a ring with identity with $|R|=p$, $p$ prime then $R$ is a field and that $R\cong \mathbb{Z}_p$ ?
Rings of Prime Cardinality
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ring-theory
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0@Andres: Because you are proving that $R$ is isomorphic as a ring to $\mathbb{Z}_p$, and $\mathbb{Z}_p$ is a field. Are you confused about why if $R$ is isomorphic to a field then it's a field? – 2012-06-19
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The additive group generated by $1$ must be the whole ring (since it is a subgroup). Thus, every element is of the form $n\cdot 1$ for some $n\in\{0,1,\ldots,p-1\}$. Moreover, $n$ is unique; the map that sends $n\cdot 1$ to $n\in\mathbb{Z}_p$ is now easily seen to be an isomorphism.
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0@Andres:Well, you have an isomorphism to $\mathbb{Z}_p$, and the conclusion holds in $\mathbb{Z}_p$. What else could you possibly need? Or, if you *must* do things the hard way, if $a\neq 0$ then $aR=R$ (must be a subgroup, and is not $0$), so there exists $b\in R$ such that $ab=1$. And likewise, $Ra=R$, so there exists $c\in R$ such that $ca=1$. And $c = c1 = c(ab) = (ca)b = 1b = b$. – 2012-06-19