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Here are two ways to solve your first problem.

• use the Python version of the complex imaginary unit, 1j.

  sage: R.<x,y> = CC['x,y'] sage: f = x + 1jy sage: f x + 1.00000000000000I*y sage: f.parent() Multivariate Polynomial Ring in x, y over Complex Field with 53 bits of precision

• redefine I to be the imaginary unit in CC.

  sage: I = CC.gen() sage: I = CC.gen() sage: f = x + I * y sage: f x + 1.00000000000000Iy sage: f.parent() Multivariate Polynomial Ring in x, y over Complex Field with 53 bits of precision

Regarding your second problem, the following fails because R is not a principal ideal domain.

sage: span([f], R)

Here are two ways to solve your first problem.

• use the Python version of the complex imaginary unit, 1j.

  sage: R.<x,y> = CC['x,y']
  sage:  f = x + 1jy
  1j*y
  sage: f
  x + 1.00000000000000I*y
  1.00000000000000*I*y
  sage: f.parent()
  Multivariate Polynomial Ring in x, y over Complex Field with 53 bits of precision
  precision
  

• redefine I to be the imaginary unit in CC.

  sage: I = CC.gen()
  sage: I = CC.gen()
  sage: f = x + I * y
  sage: f
  x + 1.00000000000000Iy
  1.00000000000000*I*y
  sage: f.parent()
  Multivariate Polynomial Ring in x, y over Complex Field with 53 bits of precision
  precision
  

Regarding your second problem, the following fails because R is not a principal ideal domain.

sage: span([f], R)