On the Weak Grothendieck Group of a Morphic Ring and its Representations

Sorokin O.S

doi:10.13189/ms.2015.030104

Abstract

The K-theoretical aspect of the commutative mophic rings is established using the arithmetical properties of the morphic rings in order to obtain a ring of all Smith normal forms of matrices over the morphic ring. The internal structure and basic properties of such rings are discussed as well as their presentations by the Witt vectors. In a case of a commutative von Neumann regular rings the famous Grothendieck group K0(R) obtains the alternative description.

Highlights

In [10] it is proved that for any element a of a commutative morphic ring R there is an element b ∈ R such that the ideals aR, bR coincides with the annihilators Ann(b), Ann(a)
One can make a partition on the pairs of the set of all principal ideals such that any element of such of this partition is uniquely determined by some pair of principal ideals
We will construct an analogue of the Grothendieck group K0(R) of a ring R using the principal ideals instead of the finitely generated projective R-modules

Summary

Introduction

In [10] it is proved that for any element a of a commutative morphic ring R there is an element b ∈ R such that the ideals aR, bR coincides with the annihilators Ann(b), Ann(a). We will construct an analogue of the Grothendieck group K0(R) of a ring R using the principal ideals instead of the finitely generated projective R-modules. Such abelian group, that is denoted as K0′ (R) and is called a weak Grothendieck group of a ring R, becomes a ring if we define a product of two elements of this group using the tensor product of principal ideals. The main motivation of these investigations is that in [11]it is proved that a commutative Bezout domain is an elementary divisor ring if and only if any quotient ring R/aR is so, where a is an arbitrary nonzero element of R. Since any finite homomorphic image of a commutative Bezout domain R is a morphic ring [14] the studies of the ring K0′ (R/aR) become related to the famous elementary divisor ring problem [8]

Preliminaries

Weak Grothendieck group