code: def test_groebner_extendend():
      M = (
          QQ.old_poly_ring(x, y, z)
          .free_module(3)
          .submodule([x + 1, y, 1], [x * y, z, z**2])
      )
      G, R = M._groebner_vec(extended=True)
      for i, g in enumerate(G):
          assert g == sum(c * gen for c, gen in zip(R[i], M.gens))
  
imports: from sympy.polys.agca.modules import FreeModule, ModuleOrder, FreeModulePolyRing
  from sympy.polys import CoercionFailed, QQ, lex, grlex, ilex, ZZ
  from sympy.abc import x, y, z
  from sympy.testing.pytest import raises
  from sympy.core.numbers import Rational
label_desc: Use image node_id 16 for calling a global function with example usage: test_groebner_extendend() without return types