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)

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

additive_order()

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

lift()

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

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

vector()

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