Let $R=\Bbb Z(i)$ be the ring of gaussian integers. Describe the cosets of the factor ring $R$ \ $A$ where $A=Ri$.

Problem: Let $$R=\Bbb Z(i)$$ be the ring of gaussian integers

Describe the cosets of the factor ring $$R$$ \ $$A$$ where $$A=Ri$$.

Thoughts:The elements of $$R$$ \ $$A$$ will be of the form: $$(a+bi)+Ri$$. I'm not really sure how to derive some sort of equivalence classes/cosets out of this. Any insights appreciated.

• Hint: $i$ is a unit of $R$. Feb 2, 2019 at 20:54

Observe that $$A=iR=R$$ and so $$R/A=\{A\}$$.
Let $$a+bi+A\in R/A$$, then $$a+bi+A=a+bi+i(-b+ai)+A=a+bi-bi-a+A=0+A=A$$