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

Question

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.

Answer 1

Observe that $A=iR=R$ and so $R/A=\{A\}$.

Let $a+bi+A\in R/A$, then $$a+bi+A=a+bi+i(-b+ai)+A=a+bi-bi-a+A=0+A=A$$