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Problem 45 (A Homomorphism From Z To A Ring With Unity)

Let $R$ be a ring with unity 1. After you have finished the first step below, you should be able to give answers to the remaining parts by referring to the first part and using the properties of ring homomorphisms.

  1. Show that the mapping $\phi:\mathbb{Z}\to R$ given by $\phi(n)=n\cdot 1$ is a ring homomorphism.
  2. Show that if $R$ has characteristic $n>0$, then $R$ contains a subring isomorphic to $Z_n$.
  3. Show that if $R$ has characteristic $n=0$, then $R$ contains a subring that is isomorphic to $Z$.
  4. Show that for any positive integer $m$, the mapping of $\phi:\mathbb{Z}\to \mathbb{Z}_m$ given by $x\to x$ mod $m$ is a ring homomorphism.

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