prove that the quotient ring S3/T3 is isomorphic to D3

Question

Could you please help with this question?

I've already shown that T_3 is an ideal of S_3.

Thanks,

Answer 1

The key idea in this case is to construct a (almost canonical) homomorphism from $S_3$ to $D_3$, whose kernel is exactly $T_3$. Existence of such a map proves two things - that $T_3$ is an ideal of $S_3$ (since it is the kernel of a ring homomorphism), and the fact that $S_3 /T_3 \cong D_3$ (from Ist Isomorphism Theorem).

Define $f : S_3 \to D_3$ by, $\begin{pmatrix} a & b & c \\ 0 & d & e \\ 0 & 0 & f \end{pmatrix} \mapsto \begin{pmatrix} a & 0 & 0 \\ 0 & d & 0 \\ 0 & 0 & f \end{pmatrix} $

One can easily check that this indeed is a ring homomorphism with kernel being precisely $T_3$.