@misc{10498/31474,
year = {2024},
month = {3},
url = {http://hdl.handle.net/10498/31474},
abstract = {We provide two new formulations of the separativity problem.
First, it is known that separativity (and strong separativity) in von Neumann regular (and exchange) rings is tightly connected to unit-regularity of certain kinds of elements. By refining this information, we characterize separative regular rings in terms of a special type of inner inverse operation, which is defined via a single identity. This shows that the separative regular rings form a subvariety of the regular rings. The separativity problem reduces to the question of whether every element of the form (1 − aa′)bac(1 − a′a) in a regular ring is unit-regular, where a′ is an inner inverse for a.
Second, it is known that separativity in exchange rings is equivalent to regular matrices over corner rings being reducible to diagonal matrices via elementary row and column operations. We show that for 2 × 2 invertible matrices, four elementary operations are sufficient, and in general also necessary. Dropping the invertibility hypothesis, but specializing to separative regular rings, we show that three elementary row operations together with three elementary column operations are sufficient, and again in general also necessary. The separativity problem can subsequently be reframed in terms of the explicit number of operations needed to diagonally reduce.},
organization = {The third author thanks the Mathematics Department of the Universitat Autònoma de Barcelona for their hospitality during his visit. The first, fourth and fifth authors were partially supported by the Spanish State Research Agency (grant No. PID2020-113047GB-I00/AEI/10.13039/501100011033). The first and fifth authors were partially supported by the Comissionat per Universitats i Recerca de la Generalitat de Catalunya (grant No. 2017-SGR-1725) and by the Spanish State Research Agency through the Severo Ochoa and María de Maeztu Program for Centers and Units of Excellence in R&D (CEX2020-001084-M). The fourth author was partially supported by PAI III grant FQM-298 of the Junta de Andalucía, and by the grant “Operator Theory: an interdisciplinary approach”, reference ProyExcel 00780, a project financed in the 2021 call for Grants for Excellence Projects, under a competitive bidding regime, aimed at entities qualified as Agents of the Andalusian Knowledge System, in the scope of the Plan Andaluz de Investigaci ón, Desarrollo e Innovación (PAIDI 2020), Consejería de Universidad, Investigación e Innovación of the Junta de Andalucía. This work was partially supported by a grant from the Simons Foundation (#963435 to Pace P. Nielsen).},
publisher = {arXiv},
keywords = {diagonally reduce},
keywords = {exchange rings},
keywords = {inner inverses},
keywords = {regular rings},
keywords = {separativity},
title = {The separativity problem in terms of varieties and diagonal reduction},
author = {Ara, Pere and Goodearl, K.R. and Nielsen, Pace and Pardo Espino, Enrique and Perera, Francesc},
}