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The problem studied in this paper is to determine some conditions on a matrix A over a ring R which will insure that the matrix equation Au = f is solvable over R if it is solvable over the residue field {A figure is presented}({A figure is presented}) for every {A figure is presented} ∈ Spec R. If R is a regular local ring (containing a field), a polynomial ring over an algebraically close field, or the ring of holomorphic functions on a Stein manifold, then a sufficient condition on A for pointwise solvability to imply global solvability is that A be generic, a concept which is defined in the paper. For the rings of functions, pointwise solvability will mean solvability over R/M for a certain set of maximal ideals. The relationship between this notion of pointwise solvability and solvability over {A figure is presented}({A figure is presented}) for all prime ideals is studied by introducing various types of closure operations on submodules. Mather has previously proved a theorem similar to the main result of this paper for the case of rings of smooth real valued functions on open subsets of Euclidean space. © 1985.

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Linear Algebra and Its Applications

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