Standard Constructions

Given one or more existing modules, various standard constructions are available to construct new modules.

Changing the Coefficient Ring

ChangeRing(M, S): ModRng, Rng -> ModRng, Map

Given a module \(M\) with base ring \(R\), together with a ring \(S\), construct the module \(N\) with base ring \(S\) obtained by coercing the components of elements of \(M\) into \(N\), together with the homomorphism from \(M\) to \(N\).

ChangeRing(M, S, f): ModRng, Rng, Map -> ModRng, Map

Given a module \(M\) with base ring \(R\), together with a ring \(S\), and a homomorphism \(f: R \rightarrow S\), construct the module \(N\) with base ring \(S\) obtained by mapping the components of elements of \(M\) into \(N\) by \(f\), together with the homomorphism from \(M\) to \(N\).

ChangeUniverse(~x, R): ModTupRngElt, Rng -> ModRng, Map

Change the coefficient ring of \(x\) to be \(R\).

Direct Sums

DirectSum(M, N): ModRng, ModRng -> ModRng, Map, Map, Map, Map

Given \(R\)-modules \(M\) and \(N\), construct the direct sum \(D\) of \(M\) and \(N\) as an \(R\)-module. The embedding maps from \(M\) into \(D\) and from \(N\) into \(D\) respectively, and the projection maps from \(D\) onto \(M\) and from \(D\) onto \(N\) respectively are also returned.

DirectSum(Q): [ ModRng ] -> [ ModRng ], [ Map ], [ Map ]

Given a sequence \(Q\) of \(R\)-modules, construct the direct sum \(D\) of these modules. The embedding maps from each of the elements of \(Q\) into \(D\) and the projection maps from \(D\) onto each of the elements of \(Q\) are also returned.