I can't understand this :if 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 this and I feel they should be of form:$0,3-i,i+(3-i),(3-i)+i(3-i)$.because the cosets of ring will be like :$\langle3-i\rangle$+$\mathbb Z[i]$.
please help Where am I wrong?
The zero element of the quotient ring $R/I$ (where $R$ is a ring and $I$ is an ideal of $R$) is simply $I$. Here, it means all multiples of $3-i$. In other words, $\{z(3-i)\mid z \in \mathbf{Z}[i]\}$.