# Changing Rings

## `ChangeRing(A, S): AlgMatV, Rng -> AlgMat, Map`

Given a matrix algebra $A$ with base ring $R$, together with a ring $S$, construct the matrix algebra $B$ with base ring $S$ obtained by coercing the components of elements of $A$ into $S$, together with the homomorphism from $A$ to $B$.

## `ChangeRing(A, S, f): AlgMatV, Rng, Map -> AlgMat, Map`

Given a matrix algebra $A$ with base ring $R$, together with a ring $S$ and a homomorphism $f: R \rightarrow S$, construct the matrix algebra $B$ with base ring $S$ obtained by mapping the components of elements of $R$ into $S$ by $f$, together with the homomorphism from $A$ to $B$.

## `hom< A -> B | f >: AlgMat, AlgMat, Map -> Map`

Given *full* matrix algebras $A$ and $B$, together with a homomorphism $f$ from the base ring $R$ of $A$ to the base ring $S$ of $B$, create the homomorphism from $A$ to $B$ which, given a matrix in $A$, applies $f$ to the entries of the matrix.