Problem 10. Consider the following map between Gaussian integers and Z₂: 6: Z[i]→ Z₂, (a + bi) = a +bmod 2. Explain why is this a ring homormorphism and show that the kernel is the ideal I generated by 1 + i, I = (1 + i). Conclude that I is a maximal ideal.

Recall that a finitely-generated ideal I is written I = (x1 , x2 , . . . , xn) where the xi ∈ R aresuch that every y ∈ I can be written y = r1x1 + · · · + rnxn for suitable choices of ri ∈ R.(a) Let R = Z and I = (4, 6). Show that I is principal and is generated by 2 ∈ Z.(b) Do the same for I = (6, 9, 15). Find a generator.(c) Now let R = Z5[x] and I = (x2 + 1, x5 + 2). Is I a principal ideal? If so, find a generator.