MatrixCombine(lrc, R)

MatrixCombine(lrc, R, lm)

Consider a polynomial ring with three variables

Consider the following four regular chains of R

Consider the following four matrices over R

We view each matrix as a result obtained modulo the corresponding regular chain in the given order. We combine these four results as follows

The four cases cannot be combined into a single one. In fact, we obtained the following two cases

The two ideals generated by rc1 and rc2 are obviously relatively prime (no common roots in z) so the Chinese Remaindering Theorem applies. However, if we try to recombine them, we create a polynomial in y with a zero-divisor as initial. This is forbidden by the properties of a regular chain.

Equations, EquiprojectableDecomposition, ExtendedRegularGcd, Matrix, PolynomialRing, RegularChains, Triangularize