Linear Algebra Over a Ring

TL;DR

This paper develops a formal matrix calculus over rings, generalizing linear algebra and providing new presentations and homological interpretations of the Grothendieck group of finitely presented modules, with applications to fields.

Contribution

Introduces a formal matrix calculus over rings, unifies linear algebra over rings with module theory, and offers new homological and algebraic insights into the Grothendieck group.

Findings

Formal matrix calculus subsumes homogeneous systems over rings.

Provides two new presentations for the Grothendieck group K_0(R-mod).

Computes the first homology group for fields, relating it to the abelianization of the multiplicative group.

Abstract

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 $K_0 (R-mod, \oplus)$ 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 $K_0 (R-mod, \oplus).$ A computation shows that if R = k is a field, then the 1-dimensional homology group is given by the abelianization of the multiplicative group of k, modulo the subgroup {1, -1}. 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.