documentation

Author: Xavier Caruso

src/doc/en/reference/oremodules/index.rst

@@ -9,17 +9,28 @@ that is an additive map satisfying the following axiom
 
     \partial(x y) = \theta(x) \partial(y) + \partial(x) y
 
-A Ore module over `(R, \theta, \partial)` is a `R`-module `M`
-equipped with a additive `f : M \to M` such that
+The Ore polynomial ring associated to these data is 
+`\mathcal S = R[X; \theta, \partial]`; its elements are the
+usual polynomials over `R` but the multiplication is twisted
+according to the rule
 
 .. MATH::
 
-    f(a x) = \theta(a) f(x) + \partial(a) x
+    \partial(x y) = \theta(x) \partial(y) + \partial(x) y
+
+We refer to :mod:`sage.rings.polynomial.ore_polynomial_ring.OrePolynomial`
+for more details.
+
+A Ore module over `(R, \theta, \partial)` is by definition a
+module over `\mathcal S`; it is the same than a `R`-module `M`
+equipped with an additive `f : M \to M` such that
 
-Such a map `f` is called a pseudomorphism.
+.. MATH::
+
+    f(a x) = \theta(a) f(x) + \partial(a) x
 
-Equivalently, a Ore module is a module over the (noncommutative)
-Ore polynomial ring `\mathcal S = R[X; \theta, \partial]`.
+Such a map `f` is called a pseudomorphism
+(see also :meth:`sage.modules.free_module.FreeModule_generic.pseudohom`).
 
 SageMath provides support for creating and manipulating Ore
 modules that are finite free over the base ring `R`.

src/sage/modules/ore_module.py

@@ -54,7 +54,7 @@
     sage: X*v
     (3*z^2 + 2*z, 2*z^2 + 4*z + 4)
 
-The method :meth:`sage.rings.modules.ore_module.OreModule.pseudohom`
+The method :meth:`sage.modules.ore_module.OreModule.pseudohom`
 returns the map `f` defining the action of `X`::
 
     sage: M.pseudohom()
@@ -162,7 +162,7 @@
 .. RUBRIC:: Morphisms of Ore modules
 
 For a tutorial on morphisms of Ore modules, we refer to
-:mod:`sage.modules.ore_modules_morphism`.
+:mod:`sage.modules.ore_module_morphism`.
 
 AUTHOR:
 
@@ -1172,7 +1172,7 @@ def identity_morphism(self):
     def _span(self, gens):
         r"""
         Return a matrix whose lines form a basis over the base field
-        of the submodule of this Ore modules generated over the Ore
+        of the submodule of this Ore module generated over the Ore
         ring by ``gens``.
 
         INPUT:

src/sage/modules/ore_module_morphism.py

@@ -124,7 +124,7 @@
     True
 
 Actually, in our use case, one can, more simply, use the method
-:meth:`sage.modules.ore_modules_morphism.OreModuleMorphism.is_injective`::
+:meth:`sage.modules.ore_module_morphism.OreModuleMorphism.is_injective`::
 
     sage: f.is_injective()
     True