# Linear Algebra Over a Ring - Mathematics > K-Theory and Homology

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Abstract: Given an associative, not necessarily commutative, ring R with identity, aformal matrix calculus is introduced and developed for pairs of matrices overR. This calculus subsumes the theory of homogeneous systems of linear equationswith coefficients in R. In the case when the ring R is a field, every pair isequivalent to a homogeneous system.Using the formal matrix calculus, two alternate presentations are given forthe Grothendieck group $K 0 R-mod, \oplus$ of the category R-mod of finitelypresented modules. One of these presentations suggests a homologicalinterpretation, and so a complex is introduced whose 0-dimensional homology isnaturally isomorphic to $K 0 R-mod, \oplus.$ A computation shows that if R =k is a field, then the 1-dimensional homology group is given by theabelianization of the multiplicative group of k, modulo the subgroup {1 -1}.The formal matrix calculus, which consists of three rules of matrixoperation, is the syntax of a deductive system whose completeness was proved byPrest. The three rules of inference of this deductive system correspond to thethree rules of matrix operation, which appear in the formal matrix calculus asthe Rules of Divisibility.

Author: ** Ivo Herzog**

Source: https://arxiv.org/