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r""" 

Elements of finitely generated modules over a PID 

 

AUTHOR: 

- William Stein, 2009 

""" 

 

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

# Copyright (C) 2009 William Stein <wstein@gmail.com> 

# 

# This program is free software: you can redistribute it and/or modify 

# it under the terms of the GNU General Public License as published by 

# the Free Software Foundation, either version 2 of the License, or 

# (at your option) any later version. 

# http://www.gnu.org/licenses/ 

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

 

from sage.structure.element import ModuleElement 

from sage.structure.richcmp import richcmp 

 

# This adds extra maybe-not-necessary checks in the code, but could 

# slow things down. It can impact what happens in more than just this 

# file. 

DEBUG=True 

 

 

class FGP_Element(ModuleElement): 

""" 

An element of a finitely generated module over a PID. 

 

INPUT: 

 

- ``parent`` -- parent module M 

 

- ``x`` -- element of M.V() 

 

EXAMPLES:: 

 

sage: V = span([[1/2,1,1],[3/2,2,1],[0,0,1]],ZZ); W = V.span([2*V.0+4*V.1, 9*V.0+12*V.1, 4*V.2]) 

sage: Q = V/W 

sage: x = Q(V.0-V.1); x #indirect doctest 

(0, 3) 

sage: isinstance(x, sage.modules.fg_pid.fgp_element.FGP_Element) 

True 

sage: type(x) 

<class 'sage.modules.fg_pid.fgp_module.FGP_Module_class_with_category.element_class'> 

sage: x is Q(x) 

True 

sage: x.parent() is Q 

True 

 

TESTS:: 

 

sage: V = span([[1/2,1,1],[3/2,2,1],[0,0,1]],ZZ); W = V.span([2*V.0+4*V.1, 9*V.0+12*V.1, 4*V.2]); Q = V/W 

sage: loads(dumps(Q.0)) == Q.0 

True 

""" 

def __init__(self, parent, x, check=DEBUG): 

""" 

INPUT: 

 

- ``parent`` -- parent module M 

 

- ``x`` -- element of M.V() 

 

- ``check`` -- (default: True) if True, verify that x in M.V() 

 

EXAMPLES:: 

 

sage: V = span([[1/2,1,1],[3/2,2,1],[0,0,1]],ZZ); W = V.span([2*V.0+4*V.1, 9*V.0+12*V.1, 4*V.2]) 

sage: Q = V/W 

sage: x = Q(V.0-V.1); type(x) 

<class 'sage.modules.fg_pid.fgp_module.FGP_Module_class_with_category.element_class'> 

sage: isinstance(x,sage.modules.fg_pid.fgp_element.FGP_Element) 

True 

 

For full documentation, see :class:`FGP_Element`. 

""" 

if check: assert x in parent.V(), 'The argument x='+str(x)+' is not in the covering module!' 

ModuleElement.__init__(self, parent) 

self._x = x 

 

def lift(self): 

""" 

Lift self to an element of V, where the parent of self is the quotient module V/W. 

 

EXAMPLES:: 

 

sage: V = span([[1/2,0,0],[3/2,2,1],[0,0,1]],ZZ); W = V.span([2*V.0+4*V.1, 9*V.0+12*V.1, 4*V.2]) 

sage: Q = V/W; Q 

Finitely generated module V/W over Integer Ring with invariants (4, 12) 

sage: Q.0 

(1, 0) 

sage: Q.1 

(0, 1) 

sage: Q.0.lift() 

(0, 0, 1) 

sage: Q.1.lift() 

(0, 2, 0) 

sage: x = Q(V.0); x 

(0, 4) 

sage: x.lift() 

(1/2, 0, 0) 

sage: x == 4*Q.1 

True 

sage: x.lift().parent() == V 

True 

 

A silly version of the integers modulo 100:: 

 

sage: A = (ZZ^1)/span([[100]], ZZ); A 

Finitely generated module V/W over Integer Ring with invariants (100) 

sage: x = A([5]); x 

doctest:...: DeprecationWarning: The default behaviour changed! 

If you *really* want a linear combination of smith generators, 

use .linear_combination_of_smith_form_gens. 

See http://trac.sagemath.org/16261 for details. 

(5) 

sage: v = x.lift(); v 

(5) 

sage: v.parent() 

Ambient free module of rank 1 over the principal ideal domain Integer Ring 

""" 

return self._x 

 

 

def __neg__(self): 

""" 

EXAMPLES:: 

 

sage: V1 = ZZ^2; W1 = V1.span([[1,2],[3,4]]); A1 = V1/W1; A1 

Finitely generated module V/W over Integer Ring with invariants (2) 

sage: -A1.0 

(1) 

sage: -A1.0 == A1.0 # order 2 

True 

""" 

P = self.parent() 

return P.element_class(P, -self._x) 

 

 

def _add_(self, other): 

""" 

EXAMPLES:: 

 

sage: V = span([[1/2,1,1],[3/2,2,1],[0,0,1]],ZZ); W = V.span([2*V.0+4*V.1, 9*V.0+12*V.1, 4*V.2]) 

sage: Q = V/W; Q 

Finitely generated module V/W over Integer Ring with invariants (4, 12) 

sage: x = Q.0; x 

(1, 0) 

sage: y = Q.1; y 

(0, 1) 

sage: x + y # indirect doctest 

(1, 1) 

sage: x + x + x + x 

(0, 0) 

sage: x + 0 

(1, 0) 

sage: 0 + x 

(1, 0) 

 

We test canonical coercion from V and W. 

 

sage: Q.0 + V.0 

(1, 4) 

sage: V.0 + Q.0 

(1, 4) 

sage: W.0 + Q.0 

(1, 0) 

sage: W.0 + Q.0 == Q.0 

True 

""" 

P = self.parent() 

return P.element_class(P, self._x + other._x) 

 

 

def _sub_(self, other): 

""" 

EXAMPLES:: 

 

sage: V = span([[1/2,1,1],[3/2,2,1],[0,0,1]],ZZ); W = V.span([2*V.0+4*V.1, 9*V.0+12*V.1, 4*V.2]) 

sage: Q = V/W; Q 

Finitely generated module V/W over Integer Ring with invariants (4, 12) 

sage: x = Q.0; x 

(1, 0) 

sage: y = Q.1; y 

(0, 1) 

sage: x - y # indirect doctest 

(1, 11) 

sage: x - x 

(0, 0) 

""" 

P = self.parent() 

return P.element_class(P, self._x - other._x) 

 

 

def _rmul_(self, c): 

""" 

Multiplication by a scalar from the left (``self`` is on the right). 

 

INPUT: 

 

- ``c`` -- an element of ``self.parent().base_ring()``. 

 

OUTPUT: 

 

The product ``c * self`` as a new instance of a module 

element. 

 

EXAMPLES:: 

 

sage: V = span([[1/2,1,1],[3/2,2,1],[0,0,1]],ZZ); W = V.span([2*V.0+4*V.1, 9*V.0+12*V.1, 4*V.2]) 

sage: Q = V/W; Q 

Finitely generated module V/W over Integer Ring with invariants (4, 12) 

sage: x = Q.0; x 

(1, 0) 

sage: 2 * x # indirect doctest 

(2, 0) 

sage: x._rmul_(4) 

(0, 0) 

sage: V = V.base_extend(QQ); W = V.span([2*V.0+4*V.1]) 

sage: Q = V/W; Q 

Vector space quotient V/W of dimension 2 over Rational Field where 

V: Vector space of degree 3 and dimension 3 over Rational Field 

Basis matrix: 

[1 0 0] 

[0 1 0] 

[0 0 1] 

W: Vector space of degree 3 and dimension 1 over Rational Field 

Basis matrix: 

[1 2 0] 

sage: x = Q.0; x 

(1, 0) 

sage: (1/2) * x # indirect doctest 

(1/2, 0) 

sage: x._rmul_(1/4) 

(1/4, 0) 

""" 

# print "_rmul_" 

P = self.parent() 

return P.element_class(P, self._x._rmul_(c)) 

 

def _lmul_(self, s): 

""" 

Multiplication by a scalar from the right (``self`` is on the left). 

 

INPUT: 

 

- ``c`` -- an element of ``self.parent().base_ring()``. 

 

OUTPUT: 

 

The product ``self * c`` as a new instance of a module 

element. 

 

EXAMPLES:: 

 

sage: V = span([[1/2,1,1],[3/2,2,1],[0,0,1]],ZZ); W = V.span([2*V.0+4*V.1, 9*V.0+12*V.1, 4*V.2]) 

sage: Q = V/W; Q 

Finitely generated module V/W over Integer Ring with invariants (4, 12) 

sage: x = Q.0; x 

(1, 0) 

sage: x * 2 # indirect doctest 

(2, 0) 

sage: x._lmul_(4) 

(0, 0) 

sage: V = V.base_extend(QQ); W = V.span([2*V.0+4*V.1]) 

sage: Q = V/W; Q 

Vector space quotient V/W of dimension 2 over Rational Field where 

V: Vector space of degree 3 and dimension 3 over Rational Field 

Basis matrix: 

[1 0 0] 

[0 1 0] 

[0 0 1] 

W: Vector space of degree 3 and dimension 1 over Rational Field 

Basis matrix: 

[1 2 0] 

sage: x = Q.0; x 

(1, 0) 

sage: x * (1/2) # indirect doctest 

(1/2, 0) 

sage: x._lmul_(1/4) 

(1/4, 0) 

""" 

# print '_lmul_' 

P = self.parent() 

return P.element_class(P, self._x._lmul_(s)) 

 

 

def _repr_(self): 

""" 

 

EXAMPLES:: 

 

sage: V = span([[1/2,1,1],[3/2,2,1],[0,0,1]],ZZ); W = V.span([2*V.0+4*V.1, 9*V.0+12*V.1, 4*V.2]) 

sage: Q = V/W 

sage: Q(V.1)._repr_() 

'(0, 1)' 

""" 

return repr(self.vector()) 

 

 

def __getitem__(self, *args): 

""" 

EXAMPLES:: 

 

sage: V = span([[1/2,0,0],[3/2,2,1],[0,0,1]],ZZ); W = V.span([2*V.0+4*V.1, 9*V.0+12*V.1, 4*V.2]) 

sage: Q = V/W; Q 

Finitely generated module V/W over Integer Ring with invariants (4, 12) 

sage: x = Q.0 + 3*Q.1; x 

(1, 3) 

sage: x[0] 

1 

sage: x[1] 

3 

sage: x[-1] 

3 

""" 

return self.vector().__getitem__(*args) 

 

def vector(self): 

""" 

EXAMPLES:: 

 

sage: V = span([[1/2,0,0],[3/2,2,1],[0,0,1]],ZZ); W = V.span([2*V.0+4*V.1, 9*V.0+12*V.1, 4*V.2]) 

sage: Q = V/W; Q 

Finitely generated module V/W over Integer Ring with invariants (4, 12) 

sage: x = Q.0 + 3*Q.1; x 

(1, 3) 

sage: x.vector() 

(1, 3) 

sage: tuple(x) 

(1, 3) 

sage: list(x) 

[1, 3] 

sage: x.vector().parent() 

Ambient free module of rank 2 over the principal ideal domain Integer Ring 

""" 

try: return self.__vector 

except AttributeError: 

self.__vector = self.parent().coordinate_vector(self, reduce=True) 

self.__vector.set_immutable() 

return self.__vector 

 

def __hash__(self): 

r""" 

TESTS:: 

 

sage: V = span([[1/2,0,0],[3/2,2,1],[0,0,1]],ZZ) 

sage: W = V.span([2*V.0+4*V.1, 9*V.0+12*V.1, 4*V.2]) 

sage: Q = V/W 

sage: x = Q.0 + 3*Q.1 

sage: hash(x) 

3713081631933328131 # 64-bit 

1298787075 # 32-bit 

 

sage: A = AdditiveAbelianGroup([3]) 

sage: hash(A.an_element()) 

3430019387558 # 64-bit 

-1659481946 # 32-bit 

""" 

return hash(self.vector()) 

 

def _vector_(self, base_ring=None): 

""" 

Support for conversion to vectors. 

 

INPUT: 

 

- ``base_ring`` -- the desired base ring of the vector. 

 

OUTPUT: 

 

A vector over the base ring. 

 

EXAMPLES:: 

 

sage: V = span([[1/2,0,0],[3/2,2,1],[0,0,1]],ZZ); W = V.span([2*V.0+4*V.1, 9*V.0+12*V.1, 4*V.2]) 

sage: Q = V/W; Q 

Finitely generated module V/W over Integer Ring with invariants (4, 12) 

sage: x = Q.0 + 3*Q.1 

sage: vector(x) 

(1, 3) 

sage: vector(CDF, x) 

(1.0, 3.0) 

 

TESTS:: 

 

sage: V = span([[1/2,0,0],[3/2,2,1],[0,0,1]],ZZ) 

sage: W = V.span([2*V.0+4*V.1, 9*V.0+12*V.1, 4*V.2]) 

sage: Q = V/W 

sage: x = Q.0 + 3*Q.1 

sage: vector(x).is_mutable() 

True 

sage: vector(CDF,x).is_mutable() 

True 

""" 

v = self.vector() 

if base_ring is None or v.base_ring() is base_ring: 

return v.__copy__() 

else: 

return v.change_ring(base_ring) 

 

def _richcmp_(self, right, op): 

""" 

Compare self and right. 

 

EXAMPLES:: 

 

sage: V = span([[1/2,1,1],[3/2,2,1],[0,0,1]],ZZ); W = V.span([2*V.0+4*V.1, 9*V.0+12*V.1, 4*V.2]) 

sage: Q = V/W; Q 

Finitely generated module V/W over Integer Ring with invariants (4, 12) 

sage: x = Q.0; x 

(1, 0) 

sage: y = Q.1; y 

(0, 1) 

sage: x == y 

False 

sage: x == x 

True 

sage: x + x == 2*x 

True 

""" 

return richcmp(self.vector(), right.vector(), op) 

 

def additive_order(self): 

""" 

Return the additive order of this element. 

 

EXAMPLES:: 

 

sage: V = span([[1/2,1,1],[3/2,2,1],[0,0,1]],ZZ); W = V.span([2*V.0+4*V.1, 9*V.0+12*V.1, 4*V.2]) 

sage: Q = V/W; Q 

Finitely generated module V/W over Integer Ring with invariants (4, 12) 

sage: Q.0.additive_order() 

4 

sage: Q.1.additive_order() 

12 

sage: (Q.0+Q.1).additive_order() 

12 

sage: V = span([[1/2,1,1],[3/2,2,1],[0,0,1]],ZZ); W = V.span([2*V.0+4*V.1, 9*V.0+12*V.1]) 

sage: Q = V/W; Q 

Finitely generated module V/W over Integer Ring with invariants (12, 0) 

sage: Q.0.additive_order() 

12 

sage: type(Q.0.additive_order()) 

<type 'sage.rings.integer.Integer'> 

sage: Q.1.additive_order() 

+Infinity 

""" 

Q = self.parent() 

I = Q.invariants() 

v = self.vector() 

 

from sage.rings.all import infinity, Mod, Integer 

from sage.arith.all import lcm 

n = Integer(1) 

for i, a in enumerate(I): 

if a == 0: 

if v[i] != 0: 

return infinity 

else: 

n = lcm(n, Mod(v[i],a).additive_order()) 

return n