The command MatrixOverChain(A, rc, R) returns the normal form of A with respect to rc. In broad terms, this is obtained by mapping RegularChains[NormalForm] on the coefficients of A.

The result is viewed as a matrix with coefficients in the total ring of fractions of R/I where I is the saturated ideal of rc.

It is assumed that rc is strongly normalized.

This command is part of the RegularChains[MatrixTools] package, so it can be used in the form MatrixOverChain(..) only after executing the command with(RegularChains[MatrixTools]).  However, it can always be accessed through the long form of the command by using RegularChains[MatrixTools][MatrixOverChain](..).