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

\[\partial(x y) = \theta(x) \partial(y) + \partial(x) y\]

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

\[\partial(x y) = \theta(x) \partial(y) + \partial(x) y\]

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

\[f(a x) = \theta(a) f(x) + \partial(a) x\]

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

Morphisms