text: """Tests for the PolynomialRing classes. """
  
  from sympy.polys.domains import QQ, ZZ
  from sympy.polys.polyerrors import ExactQuotientFailed, CoercionFailed, NotReversible
  
  from sympy.abc import x, y
  
  from sympy.testing.pytest import raises
  
  
  def test_build_order():
      R = QQ.old_poly_ring(x, y, order=(("lex", x), ("ilex", y)))
      assert R.order((1, 5)) == ((1,), (-5,))
  
  
  def test_globalring():
      Qxy = QQ.old_frac_field(x, y)
      R = QQ.old_poly_ring(x, y)
      X = R.convert(x)
      Y = R.convert(y)
  
      assert x in R
      assert 1/x not in R
      assert 1/(1 + x) not in R
      assert Y in R
      assert X.ring == R
      assert X * (Y**2 + 1) == R.convert(x * (y**2 + 1))
      assert X * y == X * Y == R.convert(x * y) == x * Y
      assert X + y == X + Y == R.convert(x + y) == x + Y
      assert X - y == X - Y == R.convert(x - y) == x - Y
      assert X + 1 == R.convert(x + 1)
      raises(ExactQuotientFailed, lambda: X/Y)
      raises(ExactQuotientFailed, lambda: x/Y)
      raises(ExactQuotientFailed, lambda: X/y)
      assert X**2 / X == X
  
      assert R.from_GlobalPolynomialRing(ZZ.old_poly_ring(x, y).convert(x), ZZ.old_poly_ring(x, y)) == X
      assert R.from_FractionField(Qxy.convert(x), Qxy) == X
      assert R.from_FractionField(Qxy.convert(x)/y, Qxy) is None
  
      assert R._sdm_to_vector(R._vector_to_sdm([X, Y], R.order), 2) == [X, Y]
  
  
  def test_localring():
      Qxy = QQ.old_frac_field(x, y)
      R = QQ.old_poly_ring(x, y, order="ilex")
      X = R.convert(x)
      Y = R.convert(y)
  
      assert x in R
      assert 1/x not in R
      assert 1/(1 + x) in R
      assert Y in R
      assert X.ring == R
      assert X*(Y**2 + 1)/(1 + X) == R.convert(x*(y**2 + 1)/(1 + x))
      assert X*y == X*Y
      raises(ExactQuotientFailed, lambda: X/Y)
      raises(ExactQuotientFailed, lambda: x/Y)
      raises(ExactQuotientFailed, lambda: X/y)
      assert X + y == X + Y == R.convert(x + y) == x + Y
      assert X - y == X - Y == R.convert(x - y) == x - Y
      assert X + 1 == R.convert(x + 1)
      assert X**2 / X == X
  
      assert R.from_GlobalPolynomialRing(ZZ.old_poly_ring(x, y).convert(x), ZZ.old_poly_ring(x, y)) == X
      assert R.from_FractionField(Qxy.convert(x), Qxy) == X
      raises(CoercionFailed, lambda: R.from_FractionField(Qxy.convert(x)/y, Qxy))
      raises(ExactQuotientFailed, lambda: X/Y)
      raises(NotReversible, lambda: X.invert())
  
      assert R._sdm_to_vector(
          R._vector_to_sdm([X/(X + 1), Y/(1 + X*Y)], R.order), 2) == \
          [X*(1 + X*Y), Y*(1 + X)]
  
  
  def test_conversion():
      L = QQ.old_poly_ring(x, y, order="ilex")
      G = QQ.old_poly_ring(x, y)
  
      assert L.convert(x) == L.convert(G.convert(x), G)
      assert G.convert(x) == G.convert(L.convert(x), L)
      raises(CoercionFailed, lambda: G.convert(L.convert(1/(1 + x)), L))
  
  
  def test_units():
      R = QQ.old_poly_ring(x)
      assert R.is_unit(R.convert(1))
      assert R.is_unit(R.convert(2))
      assert not R.is_unit(R.convert(x))
      assert not R.is_unit(R.convert(1 + x))
  
      R = QQ.old_poly_ring(x, order='ilex')
      assert R.is_unit(R.convert(1))
      assert R.is_unit(R.convert(2))
      assert not R.is_unit(R.convert(x))
      assert R.is_unit(R.convert(1 + x))
  
      R = ZZ.old_poly_ring(x)
      assert R.is_unit(R.convert(1))
      assert not R.is_unit(R.convert(2))
      assert not R.is_unit(R.convert(x))
      assert not R.is_unit(R.convert(1 + x))
  