UTM Ideals Varieties and Algorithm - Chapter 1 Section 5 Exercise 5

Problem:

Solution

If $ p \in \langle f - qg, g \rangle $, then $ p = a(f-qg) + bg = af + (b- aq)g \in \langle f, g \rangle $

If $ p \in \langle f, g \rangle $, then $ p = af + bg = a(f-qg) + (b+ aq)g \in \langle f-qg, g \rangle $

Therefore $ \langle f - qg, g \rangle =\langle f, g \rangle $.