Internals: promote and lift

The commands promote and lift are usually for when we have a natural map of rings A --> B. In this situation, promote applies the map to an element of A, and lift computes a preimage (typically some sort of normal form.)

A = QQ[x]
B = A[y]
elem1 = y^3 -- in B
C = B/(x^2-y^3)
elem2 = 1/2*x^3 -- in A
elem3 = elem1 + elem2 -- C
lift(elem3, B) -- y+x^3 in B

When we add elem1 and elem2 as above, new methods (+, B, A) and (+, B, B) are created behind the scenes and stashed in the youngest object B as described here. Type code (symbol +, RingElement, RingElement) to see the usage of promote.