Extending and contracting maximal ideals in the function rings of pointfree topology

By first describing the fixed maximal ideals of RL and those of its bounded part, R* L, we show that every fixed maximal ideal of the bigger ring contracts to a fixed maximal ideal of the smaller ring, and every fixed maximal ideal of the smaller ring extends to a fixed maximal ideal of the bigger ring. However, the only instance where every maximal ideal of R* L extends to a maximal ideal of RL is when the two rings coincide. This allows the deduction that RL is integral over R* L if and only if L is pseudocompact.