TY - JOUR
T1 - On the equationally artinian groups
AU - Shahryari, Mohammad
AU - Tayyebi, Javad
N1 - Publisher Copyright:
© Siberian Federal University. All rights reserved.
PY - 2020
Y1 - 2020
N2 - In this article, we study the property of being equationally Artinian in groups. We define the radical topology corresponding to such groups and investigate the structure of irreducible closed sets of these topologies. We prove that a finite extension of an equationally Artinian group is again equationally Artinian. We also show that a quotient of an equationally Artinian group of the form G[t] by a normal subgroup which is a finite union of radicals, is again equationally Artnian. A necessary and sufficient condition for an Abelian group to be equationally Artinian will be given as the last result. This will provide a large class of examples of equationally Artinian groups.
AB - In this article, we study the property of being equationally Artinian in groups. We define the radical topology corresponding to such groups and investigate the structure of irreducible closed sets of these topologies. We prove that a finite extension of an equationally Artinian group is again equationally Artinian. We also show that a quotient of an equationally Artinian group of the form G[t] by a normal subgroup which is a finite union of radicals, is again equationally Artnian. A necessary and sufficient condition for an Abelian group to be equationally Artinian will be given as the last result. This will provide a large class of examples of equationally Artinian groups.
KW - Algebraic geometry over groups
KW - Equationally Artinian groups
KW - Equationally Noetherian groups
KW - Radical topology
KW - Radicals
KW - Systems of group equations
KW - Zariski topology
UR - http://www.scopus.com/inward/record.url?scp=85092084620&partnerID=8YFLogxK
UR - http://www.scopus.com/inward/citedby.url?scp=85092084620&partnerID=8YFLogxK
U2 - 10.17516/1997-1397-2020-13-5-583-595
DO - 10.17516/1997-1397-2020-13-5-583-595
M3 - Article
AN - SCOPUS:85092084620
SN - 1997-1397
VL - 13
SP - 583
EP - 595
JO - Journal of Siberian Federal University - Mathematics and Physics
JF - Journal of Siberian Federal University - Mathematics and Physics
IS - 5
ER -