Linear Algebra over a Ring

Given an associative, not necessarily commutative, ring R with identity, a formal matrix calculus is introduced and developed for pairs of matrices over R. This calculus subsumes the theory of homogeneous systems of linear equations with coefficients in R. In the case when the ring R is a field, every pair is equivalent to a homogeneous system. Using the formal matrix calculus, two alternate presentations are given for the Grothendieck group K0(R-mod,⊕) of the category R-mod of finitely presented modules. One of these presentations suggests a homological interpretation, and so a complex is introduced whose 0-dimensional homology is naturally isomorphic to K0(R-mod,⊕). A computation shows that if R = k is a field, then the 1-dimensional homology group is given by (k×)ab/{±1}, where k × denotes the multiplicitave group of k, and (k×)ab its abelianization. The formal matrix calculus, which consists of three rules of matrix operation, is the syntax of a deductive system whose completeness was proved by Prest. The three rules of inference of this deductive system correspond to the three rules of matrix operation, which appear in the formal matrix calculus as the Rules of Divisibility. Let R be an associative, not necessarily commutative, ring with identity. For every natural number n, define Ln(R) to be the collection of pairs (B | A) where A and B are matrices with entries in R such that A has n columns, and B has the same number of rows as A. Define the relation (B | A) ≤n (B ′ | A) to hold in Ln(R) provided there exist matrices U, V and G, of appropriate size, such that UB = BV and UA = A +BG. Separating out the individual roles of the three matrices, one verifies easily (Theorem 2) that this relation is the least pre-order on Ln(R) satisfying the following three Rules of (Left) Divisibility (RoD) for matrices: (1) if U is a matrix with m columns, then (B | A) ≤n (UB | UA). (2) if V is a matrix with k rows, then (BV | A) ≤n (B | A). (3) if G is a k × n matrix, then (B | A+BG) ≤n (B | A) In this article, we develop the formal matrix calculus that arises from these rules. Two pairs in Ln(R) are equivalent (B | A) ≈n (B ′ | A) provided both (B | A) ≤n (B ′ | A) and (B | A) ≤n (B | A) hold; an n-ary matrix pair [B | A] is defined to be an equivalence class of this relation. The collection of n-ary matrix pairs is denoted by Ln(R); the partial order induced on Ln(R) by ≤n is denoted using the same notation. This partial order Ln(R) of n-ary matrix pairs has a maximum element 1n that satisfies Proposition 9. [B | A] = 1n if and only if there exists a matrix W such that A = BW. 2000 Mathematics Subject Classification. 03B22, 06C05, 15A24, 16E20, 18F30, 19D55.