code: def _dmp_rr_trivial_gcd(f, g, u, K):
      """Handle trivial cases in GCD algorithm over a ring."""
      zero_f = dmp_zero_p(f, u)
      zero_g = dmp_zero_p(g, u)
      if_contain_one = dmp_one_p(f, u, K) or dmp_one_p(g, u, K)
  
      if zero_f and zero_g:
          return tuple(dmp_zeros(3, u, K))
      elif zero_f:
          if K.is_nonnegative(dmp_ground_LC(g, u, K)):
              return g, dmp_zero(u), dmp_one(u, K)
          else:
              return (
                  dmp_neg(g, u, K),
                  dmp_zero(u),
                  dmp_ground(-K.one, u),
              )
      elif zero_g:
          if K.is_nonnegative(dmp_ground_LC(f, u, K)):
              return f, dmp_one(u, K), dmp_zero(u)
          else:
              return (
                  dmp_neg(f, u, K),
                  dmp_ground(-K.one, u),
                  dmp_zero(u),
              )
      elif if_contain_one:
          return dmp_one(u, K), f, g
      elif query("USE_SIMPLIFY_GCD"):
          return _dmp_simplify_gcd(f, g, u, K)
      else:
          return None
  
imports: from sympy.polys.densearith import dup_sub_mul, dup_neg, dmp_neg, dmp_add, dmp_sub, dup_mul, dmp_mul, dmp_pow, dup_div, dmp_div, dup_rem, dup_quo, dmp_quo, dup_prem, dmp_prem, dup_mul_ground, dmp_mul_ground, dmp_mul_term, dup_quo_ground, dmp_quo_ground, dup_max_norm, dmp_max_norm
  from sympy.polys.densebasic import dup_strip, dmp_raise, dmp_zero, dmp_one, dmp_ground, dmp_one_p, dmp_zero_p, dmp_zeros, dup_degree, dmp_degree, dmp_degree_in, dup_LC, dmp_LC, dmp_ground_LC, dmp_multi_deflate, dmp_inflate, dup_convert, dmp_convert, dmp_apply_pairs
  from sympy.polys.densetools import dup_clear_denoms, dmp_clear_denoms, dup_diff, dmp_diff, dup_eval, dmp_eval, dmp_eval_in, dup_trunc, dmp_ground_trunc, dup_monic, dmp_ground_monic, dup_primitive, dmp_ground_primitive, dup_extract, dmp_ground_extract
  from sympy.polys.galoistools import gf_int, gf_crt
  from sympy.polys.polyconfig import query
  from sympy.polys.polyerrors import MultivariatePolynomialError, HeuristicGCDFailed, HomomorphismFailed, NotInvertible, DomainError
label_desc: Use image node_id 27 for calling a global function with example usage: _dmp_rr_trivial_gcd(f, g, u, K) and returns: tuple, g,  dmp_zero,  dmp_one, dmp_neg,  dmp_zero,  dmp_ground, f,  dmp_one,  dmp_zero, dmp_neg,  dmp_ground,  dmp_zero, dmp_one,  f,  g, _dmp_simplify_gcd, None