Modules over Ore rings

Let $R$ be a commutative ring, $\theta :K\to K$ by a ring endomorphism and $\partial :K\to K$ be a $\theta$-derivation, that is an additive map satisfying the following axiom

$$\partial (xy)=\theta (x)\partial (y)+\partial (x)y$$

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

$$\partial (xy)=\theta (x)\partial (y)+\partial (x)y$$

We refer to 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

$$f(ax)=\theta (a)f(x)+\partial (a)x$$

Such a map $f$ is called a pseudomorphism (see also sage.modules.free_module.FreeModule_generic.pseudohom()).

SageMath provides support for creating and manipulating Ore modules that are finite free over the base ring $R$. This includes, in particular, Frobenius modules and modules with connexions.

Modules, submodules and quotients

• Ore modules
• Elements in Ore modules

Morphisms

• Space of morphisms between Ore modules
• Morphisms between Ore modules