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We are to find number of elements and charecterisstic of $R =$ $Z[I]/\langle 2+2I \rangle$. Here $\gcd (2,2) \ne 1$ .So I can't say that it is isomorphic to $Z_8$ . By the fact that $Z[i]$ is ED I can say that $R$ has those elements which have norm $0$ to $7$.But then I have $0,1,i,-1,-i,1+i,1-i,\dots$ "so on ! Is it true? It seems horrible to me to find charecteristic from my observaation (not sure is true). So ,I need a hint , thanks for reading!

Bernard
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Subhajit Saha
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    I'm the \langle \rangle fairy, here to let you know that $\langle, \rangle$ plays nicer with TeX than <, > does :) – Patrick Stevens May 04 '19 at 07:53
  • The existence of $1$ implies that the characteristic of the quotient ring is necessarily equal to the smallest positive integer in the ideal you divide by. So what is the smallest positive integer you can find in $\langle 2+2i\rangle$? – Arthur May 04 '19 at 07:57
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    @Arthur ,four . – Subhajit Saha May 04 '19 at 08:05

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