Elements of finitely generated modules over a PID#

AUTHOR:

  • William Stein, 2009

class sage.modules.fg_pid.fgp_element.FGP_Element(parent, x, check=True)[source]#

Bases: 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, 9)
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

>>> from sage.all import *
>>> V = span([[Integer(1)/Integer(2),Integer(1),Integer(1)],[Integer(3)/Integer(2),Integer(2),Integer(1)],[Integer(0),Integer(0),Integer(1)]],ZZ); W = V.span([Integer(2)*V.gen(0)+Integer(4)*V.gen(1), Integer(9)*V.gen(0)+Integer(12)*V.gen(1), Integer(4)*V.gen(2)])
>>> Q = V/W
>>> x = Q(V.gen(0)-V.gen(1)); x #indirect doctest
(0, 9)
>>> isinstance(x, sage.modules.fg_pid.fgp_element.FGP_Element)
True
>>> type(x)
<class 'sage.modules.fg_pid.fgp_module.FGP_Module_class_with_category.element_class'>
>>> x is Q(x)
True
>>> x.parent() is Q
True
additive_order()[source]#

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())
<class 'sage.rings.integer.Integer'>
sage: Q.1.additive_order()
+Infinity

>>> from sage.all import *
>>> V = span([[Integer(1)/Integer(2),Integer(1),Integer(1)],[Integer(3)/Integer(2),Integer(2),Integer(1)],[Integer(0),Integer(0),Integer(1)]],ZZ); W = V.span([Integer(2)*V.gen(0)+Integer(4)*V.gen(1), Integer(9)*V.gen(0)+Integer(12)*V.gen(1), Integer(4)*V.gen(2)])
>>> Q = V/W; Q
Finitely generated module V/W over Integer Ring with invariants (4, 12)
>>> Q.gen(0).additive_order()
4
>>> Q.gen(1).additive_order()
12
>>> (Q.gen(0)+Q.gen(1)).additive_order()
12
>>> V = span([[Integer(1)/Integer(2),Integer(1),Integer(1)],[Integer(3)/Integer(2),Integer(2),Integer(1)],[Integer(0),Integer(0),Integer(1)]],ZZ); W = V.span([Integer(2)*V.gen(0)+Integer(4)*V.gen(1), Integer(9)*V.gen(0)+Integer(12)*V.gen(1)])
>>> Q = V/W; Q
Finitely generated module V/W over Integer Ring with invariants (12, 0)
>>> Q.gen(0).additive_order()
12
>>> type(Q.gen(0).additive_order())
<class 'sage.rings.integer.Integer'>
>>> Q.gen(1).additive_order()
+Infinity
lift()[source]#

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, 6, 1)
sage: Q.1.lift()
(0, -2, 0)
sage: x = Q(V.0); x
(0, 8)
sage: x.lift()
(1/2, 0, 0)
sage: x == 8*Q.1
True
sage: x.lift().parent() == V
True

>>> from sage.all import *
>>> V = span([[Integer(1)/Integer(2),Integer(0),Integer(0)],[Integer(3)/Integer(2),Integer(2),Integer(1)],[Integer(0),Integer(0),Integer(1)]],ZZ); W = V.span([Integer(2)*V.gen(0)+Integer(4)*V.gen(1), Integer(9)*V.gen(0)+Integer(12)*V.gen(1), Integer(4)*V.gen(2)])
>>> Q = V/W; Q
Finitely generated module V/W over Integer Ring with invariants (4, 12)
>>> Q.gen(0)
(1, 0)
>>> Q.gen(1)
(0, 1)
>>> Q.gen(0).lift()
(0, 6, 1)
>>> Q.gen(1).lift()
(0, -2, 0)
>>> x = Q(V.gen(0)); x
(0, 8)
>>> x.lift()
(1/2, 0, 0)
>>> x == Integer(8)*Q.gen(1)
True
>>> 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
(5)
sage: v = x.lift(); v
(5)
sage: v.parent()
Ambient free module of rank 1 over the principal ideal domain Integer Ring

>>> from sage.all import *
>>> A = (ZZ**Integer(1))/span([[Integer(100)]], ZZ); A
Finitely generated module V/W over Integer Ring with invariants (100)
>>> x = A([Integer(5)]); x
(5)
>>> v = x.lift(); v
(5)
>>> v.parent()
Ambient free module of rank 1 over the principal ideal domain Integer Ring
vector()[source]#

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

>>> from sage.all import *
>>> V = span([[Integer(1)/Integer(2),Integer(0),Integer(0)],[Integer(3)/Integer(2),Integer(2),Integer(1)],[Integer(0),Integer(0),Integer(1)]],ZZ); W = V.span([Integer(2)*V.gen(0)+Integer(4)*V.gen(1), Integer(9)*V.gen(0)+Integer(12)*V.gen(1), Integer(4)*V.gen(2)])
>>> Q = V/W; Q
Finitely generated module V/W over Integer Ring with invariants (4, 12)
>>> x = Q.gen(0) + Integer(3)*Q.gen(1); x
(1, 3)
>>> x.vector()
(1, 3)
>>> tuple(x)
(1, 3)
>>> list(x)
[1, 3]
>>> x.vector().parent()
Ambient free module of rank 2 over the principal ideal domain Integer Ring