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promote -- promote to another ring

Synopsis

Description

Promote the given ring element or matrix f to an element or matrix of R, via the natural map to R. This is semantically equivalent to creating the natural ring map from ring f --> R and mapping f via this map.

i1 : R = QQ[a..d]; f = a^2;
 
i3 : S = R/(a^2-b-1);
 
i4 : promote(2/3,S)

     2
o4 = -
     3

o4 : S
 
i5 : F = map(R,QQ);  F(2/3)

o5 : RingMap R <-- QQ

     2
o6 = -
     3

o6 : R
 
i7 : promote(f,S)

o7 = b + 1

o7 : S
 
i8 : G = map(S,R); G(f)

o8 : RingMap S <-- R

o9 = b + 1

o9 : S

Promotion of real numbers to rational numbers is accomplished by using all of the bits of the internal representation.

i10 : promote(101.,QQ)

o10 = 101

o10 : QQ
 
i11 : promote(.101,QQ)

       3638908498915361
o11 = -----------------
      36028797018963968

o11 : QQ
 
i12 : factor denominator oo

       55
o12 = 2

o12 : Expression of class Product
 
i13 : ooo + 0.

o13 = .101

o13 : RR (of precision 53)
 
i14 : oo === .101

o14 = true

For promotion of ring elements, there is the following shorter notation.

i15 : 13_R

o15 = 13

o15 : R

If you wish to promote a module to another ring, either promote the corresponding matrices, use the natural ring map, or use tensor product of matrices or modules.

i16 : use R;
 
i17 : I = ideal(a^2,a^3,a^4)

              2   3   4
o17 = ideal (a , a , a )

o17 : Ideal of R
 
i18 : promote(I,S)

                              2
o18 = ideal (b + 1, a*b + a, b  + 2b + 1)

o18 : Ideal of S
 
i19 : m = image matrix{{a^2,a^3,a^4}}

o19 = image | a2 a3 a4 |

                              1
o19 : R-module, submodule of R
 
i20 : promote(gens m,S)

o20 = | b+1 ab+a b2+2b+1 |

              1      3
o20 : Matrix S  <-- S
 
i21 : G m

o21 = image | b+1 ab+a b2+2b+1 |

                              1
o21 : S-module, submodule of S
 
i22 : m ** S

o22 = cokernel {2} | a  0  |
               {3} | -1 a  |
               {4} | 0  -1 |

                             3
o22 : S-module, quotient of S
A special feature is that if f is rational, and R is not an algebra over QQ, then an element of R is provided by attempting the evident division.

See also

Ways to use promote :

For the programmer

The object promote is a method function.