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))
dataset/ETHPy150Open sympy/sympy/sympy/polys/agca/tests/test_modules.py/test_groebner_extendend/original