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# 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

- Ore modules
- Elements in Ore modules

## Morphisms

- Space of morphisms between Ore modules
- Morphisms between Ore modules