Given two groups \((G, \ast)\) and \((H, \star)\text{,}\) we define a new group \(G\times H = \{(g,h) \st g \in G \text{ and } h \in H\}\) with operation defined by \((g_1, h_1) \cdot (g_2, h_2) = (g_1\ast g_2, h_1 \star h_2)\text{.}\) That is, \(G\times H\) is the set of all ordered pairs with the first coordinate from \(G\) and the second coordinate from \(H\text{.}\) The operation is done “component-wise”: use \(G\)’s operation for the first coordinate and \(H\)’s operation for the second component.
The isomorphism \(\varphi\) you found in the previous part satisfies \(\varphi(a+b) = \varphi(a)+\varphi(b)\text{.}\) Does this same function also “respect” multiplication? That is does it satisfy the isomorphism property for multiplication:
Find both non-trivial ideals of \(R = \Z_3 \times \Z_3\text{.}\) Then find a subring that is not and ideal. Explain how you know the non-ideal subring really is not an ideal (show specifically what goes wrong).
Pick one of the ideals, call it \(J\text{.}\) Call the non-ideal \(S\text{.}\) For both, write out their cosets in \(R\text{.}\) That is find the sets \(R/J\) and \(R/S\text{.}\)
Illustrate that \(R/J\) is really a ring (a quotient ring) and that \(R/S\) is not a ring. That is, show specifically what fails when you try to define the ring operations on the set of cosets \(R/S\text{.}\)
Let \(G_1\) and \(G_2\) be groups and assume that \(G_1 \cong G_2\text{,}\) with isomorphism \(\varphi:G_1 \to G_2\text{.}\) Let \(H_1\) be a subgroup of \(G_1\text{.}\) Define \(H_2 = \varphi(H_1) = \{x \in G_2 \st x = \varphi(h) \text{ for some } h \in H_1\}\text{.}\)
