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 = x^3 -- in A
methods symbol +
elem3 = elem1+elem2 -- in B
elem4 = promote(elem3,C) -- x^3+x^2 in C
elem5 = lift(elem4,B) -- in B

In the line elem3 = elem1+elem2 , 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 another use of promote in this case.