Doubt regarding zero elements in factor ring :$\mathbb Z[i]/\langle3-i\rangle$

Question

I have the factor ring $\mathbb Z[i]/\langle3-i\rangle$ and am asked to find elements zero in this ,they are $0,3-i,i(3-i),(3-i)+i(3-i)$.

But I can't understand how do we guarantee these are the only zero elements and there are no more.....

Please help I'm STRUCK.......

Answer 1

The zero element (which is always unique) of the quotient ring $R/I$ is the ideal $I$. In present question, $I$ consists of all elements $(3-i)x$ with $x\in\mathbb{Z}[i]$.