# Related Maps

This section is devoted to maps between differential operator rings.

## `TranslationMap(R, e): RngDiffOp, RngElt -> Map`

Returns a map on the differential operator ring $R$ that replaces $R.1$ by $R.1+e$ when applied to a differential operator for some suitable ring element $e$.

## `LiftMap(m, R): Map, RngDiffOp -> Map`

Let $m : F \to M$ be a differential map on differential fields and $R$ a differential operator ring over $F$. Then this routine lifts the given map to a map on the differential operator rings $R \to S$, where the basefield of $S$ is $M$.

## `Example Maps Between Diff Op Rings (ex-943ab0)`

```magma
> F<z> := RationalDifferentialField(Rationals());
> R<D> := DifferentialOperatorRing(F);
> transmap := TranslationMap(R, 2 + z);
> Codomain(transmap) eq R;
> transmap(D);
D + z + 2
> transmap(D^2);
D^2 + (2*z + 4)*D + z^2 + 4*z + 5
> P<T> := PolynomialRing(F);
> M<u>, mp := ext<F|T^2+z>;
> liftmap := LiftMap(mp, R);
> Rprime<Dprime> := Codomain(liftmap);
> IsDifferentialOperatorRing(Rprime);
true
> BaseRing(Rprime) eq M;
true
> liftmap(D);
Dprime
> liftmap(R!z);
z
> Derivation(Rprime)(liftmap(z));
1
> Derivation(Rprime)(u);
1/2/z*u

```