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Problem 25 (A Factor Ring Of The Gaussian Integers)

Let $R=\mathbb{Z}[i]$, $a=3-i$, and let $A=\left<a\right>$. We would like to examine the factor ring $R/A$, determine how many elements are in this factor ring, and find a simple ring to which $\mathbb{Z}[i]/\left<3-i\right>$ is isomorphic.

  1. Why do we know $3+A=i+A$? Use this to explain why $10+A=0+A$.
  2. Explain why every element of $R/A$ must be of the form $k+A$ where $k\in \{0,1,2,\ldots 9\}$, so there are at most 10 elements.
  3. Suppose the integer $k$ is an element of $A = \left<a\right> = \{ra\mid r\in R\}$. In other words, suppose that there exists an element $r\in R$ such that $k+0i = ra = (c+di)(3-i)$. Use this to show that $k$ must be a multiple of 10.
  4. State a much simpler ring to which $R/A$ is isomorphic, you don't need to prove your answer.

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