Coverage for local/lib/python2.7/site-packages/sage/categories/domains.py : 45%

r""" 

Domains 

""" 

#***************************************************************************** 

# Copyright (C) 2008 Teresa Gomez-Diaz (CNRS) <Teresa.Gomez-Diaz@univ-mlv.fr> 

# 2012 Nicolas M. Thiery <nthiery at users.sf.net> 

# 

# Distributed under the terms of the GNU General Public License (GPL) 

# http://www.gnu.org/licenses/ 

#****************************************************************************** 

from sage.misc.lazy_import import LazyImport 

from sage.categories.category_with_axiom import CategoryWithAxiom 

from sage.categories.rings import Rings 

class Domains(CategoryWithAxiom): 

""" 

The category of domains 

A domain (or non-commutative integral domain), is a ring, not 

necessarily commutative, with no nonzero zero divisors. 

EXAMPLES:: 

sage: C = Domains(); C 

Category of domains 

sage: C.super_categories() 

[Category of rings] 

sage: C is Rings().NoZeroDivisors() 

True 

TESTS:: 

sage: TestSuite(C).run() 

""" 

_base_category_class_and_axiom = (Rings, "NoZeroDivisors") 

def super_categories(self): 

""" 

EXAMPLES:: 

sage: Domains().super_categories() 

[Category of rings] 

""" 

return [Rings()] 

Commutative = LazyImport('sage.categories.integral_domains', 'IntegralDomains', at_startup=True) 

class ParentMethods: 

def _test_zero_divisors(self, **options): 

""" 

Check to see that there are no zero divisors. 

.. NOTE:: 

In rings whose elements can not be represented exactly, there 

may be zero divisors in practice, even though these rings do 

not have them in theory. For such inexact rings, these tests 

are not performed: 

sage: R = ZpFM(5); R 

5-adic Ring of fixed modulus 5^20 

sage: R.is_exact() 

False 

sage: a = R(5^19) 

sage: a.is_zero() 

False 

sage: (a*a).is_zero() 

True 

sage: R._test_zero_divisors() 

EXAMPLES:: 

sage: ZZ._test_zero_divisors() 

sage: ZpFM(5)._test_zero_divisors() 

""" 

if not self.is_exact(): 

return # Can't check on inexact rings 

tester = self._tester(**options) 

# Filter out zero 

S = [s for s in tester.some_elements() if not s.is_zero()] 

from sage.misc.misc import some_tuples 

for a,b in some_tuples(S, 2, tester._max_runs): 

p = a * b 

tester.assertFalse(p.is_zero()) 

class ElementMethods: 

pass