Working with multiple rings is more subtle than simply replacing values of the variables in a ring. On the other hand it is particularly easy in Macaulay2. We define a sequence of rings below and move between each to show both the dangers and the convenience. ## defining multiple rings

Notice that Macaulay2 sees the coefficient ring as R1, we could just as easily defined `R2` as `R1[s,t]` . Movement and addition between these rings is easy.

Since `I` is defined as an ideal in `R2` we cannot type `ZZ/101[s,t]/I` as the computer sees `ZZ/101[s,t]` as different from `R2` and so does not see `I` as being in this ring. For more about defining rings see rings. We now work with moving between `R2` and `R3`. ## moving between rings using use and substitute

f and g are elements in `R3` now and this is shown by the fact that Macaulay2 sees them as `-t^2` and 0. To recover these elements as polynomials in `R2` type `use R2` and define them again in `R2`. The command substitute does not work well here, where as if we want to see the image of elements of `R2` in `R3` it does work well and without using the command `use`. Macaulay2 always tells you which ring an element is in on the line after it prints the ring element.

## subtleties of substitute and describe

Now we complicate things further by constructing a fraction field and then further constructing polynomial rings and quotient rings. First we see that while describe helped us to see how we defined `R2` and `R3`, the same does not hold when a fraction field is constructed. Note that R3 is a domain.

The command substitute works well to move elements from `R2` or `R3` to `R4`. An alternative to substitute is to form the canonical injection of R3 into R4 (the same can be done for the canonical projection from R2 to R3 above - we do the example here). To move elements from `R4` back to `R3` an alternate method must be used. Also, the method of constructing a map does not work well in the reverse direction for the same reasons substitute does not.

## non-standard coefficient fields

We can go through the whole process again using R4 now as the field.

Notice that at each stage Macaulay2 only refers back to the last ring we defined. All of the methods above still work here in theory, but caution is advised. We give an example below to illustrate. Also, note that many other computations will no longer work, because GrÃ¶bner basis computations only work over ZZ, `ZZ/n` and QQ at this time. ## using maps to move between rings

Macaulay2 claims this is the zero map, and that the image of `f` is 1, but we know better. By forming a series of maps and composing them we see the map that makes sense. We also contrast the map with using `substitute`.

## elements versus matrices

Finally, note that everywhere we used the element `f` we can place a matrix or an ideal and get similar results. ## substitute(J,vars R)

We close this long example with a brief discussion of `substitute(J,vars R)`. This command is more sensitive than `substitute` as it will give an error message when the variables involved do not match up.

i1 : R1 = ZZ/101; |

i2 : R2 = ZZ/101[s,t]; |

i3 : describe R2 o3 = R1[s..t, Degrees => {2:1}, Heft => {1}, MonomialOrder => {MonomialSize => 32}, DegreeRank => 1] {GRevLex => {2:1} } {Position => Up } |

i4 : I = ideal (s^4+t^2+1); o4 : Ideal of R2 |

i5 : R3 = R2/I; |

i6 : describe R3 R2 o6 = ----------- 4 2 s + t + 1 |

i7 : f = s^4+1 2 o7 = -t o7 : R3 |

i8 : g = s^4+t^2+1 o8 = 0 o8 : R3 |

i9 : use R2; |

i10 : substitute(g,R2) o10 = 0 o10 : R2 |

i11 : f = s^4+1 4 o11 = s + 1 o11 : R2 |

i12 : g = s^4+t^2+1 4 2 o12 = s + t + 1 o12 : R2 |

i13 : substitute(f,R3) 2 o13 = -t o13 : R3 |

i14 : describe R3 R2 o14 = ----------- 4 2 s + t + 1 |

i15 : R4 = frac R3; |

i16 : describe R4 / R2 \ o16 = frac|-----------| | 4 2 | \s + t + 1/ |

i17 : use R2; |

i18 : f = s^4+1; |

i19 : substitute(f,R4) 2 o19 = -t o19 : R4 |

i20 : use R3; |

i21 : g = substitute(f,R3); |

i22 : substitute(g,R4) 2 o22 = -t o22 : R4 |

i23 : F = map(R4,R3) o23 = map(R4,R3,{s, t}) o23 : RingMap R4 <--- R3 |

i24 : F(f) 2 o24 = -t o24 : R4 |

i25 : R5 = R4[u,v,w]; |

i26 : describe R5 o26 = R4[u..w, Degrees => {3:1}, Heft => {1}, MonomialOrder => {MonomialSize => 32}, DegreeRank => 1] {GRevLex => {3:1} } {Position => Up } |

i27 : J = ideal(u^3-v^2*w+w^3,v^2+w^2,u*v-v*w+u*w) 3 2 3 2 2 o27 = ideal (u - v w + w , v + w , u*v + u*w - v*w) o27 : Ideal of R5 |

i28 : R6 = R5/J; |

i29 : describe R6 R5 o29 = ----------------------------------------- 3 2 3 2 2 (u - v w + w , v + w , u*v + u*w - v*w) |

i30 : map(R6,R2) o30 = map(R6,R2,{s, t}) o30 : RingMap R6 <--- R2 |

i31 : substitute(f,R6) 2 o31 = -t o31 : R6 |

i32 : use R2; |

i33 : f = s^4+1; |

i34 : F = map(R4,R2); o34 : RingMap R4 <--- R2 |

i35 : G = map(R5,R4); o35 : RingMap R5 <--- R4 |

i36 : H = map(R6,R5); o36 : RingMap R6 <--- R5 |

i37 : H(G(F(f))) 2 o37 = -t o37 : R6 |

i38 : f1 = substitute(f,R4) 2 o38 = -t o38 : R4 |

i39 : f2 = substitute(f1,R5) 2 o39 = -t o39 : R5 |

i40 : substitute(f2,R6) 2 o40 = -t o40 : R6 |

i41 : substitute(f,vars R3) 2 o41 = -t o41 : R3 |

i42 : try substitute(f,vars R5) else "found error" o42 = found error |