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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2**Hint** Consider the image of $\,\mathbb Z\,$ in $R$, i.e. the *characteristic* subring. – 2012-06-19
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0I applied a permutation on the title so it would make more sense (methinks). – 2012-06-19
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0@Asaf: An unfortunately consequence of the fact that language is not commutative... – 2012-06-19
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0Prove that for any ring, the elements of the form $1+1+\dots +1$ form a subring subring of $R$, isomorphic to a $\mathbb{Z}_n$. Since the order of a subring must divide the order of the ring, we know that this ring is in fact the whole ring $R$. – 2012-06-19
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0Lieven and as it follows that R is a field? – 2012-06-19
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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