Geodesic
On the equational Artinian algebras
Sibirskie èlektronnye matematičeskie izvestiâ, Tome 13 (2016), pp. 875-881.

Voir la notice de l'article provenant de la source Math-Net.Ru

Equational Artinian algebras were introduced in our previous work: Compactness conditions in universal algebraic geometry, Algebra and Logic, 2016, 55 (2). In this note, we define the notion of radical topology with respect to an algebra $A$ and using the well-known König lemma in graph theory, we show that the algebra $A$ is equational Artinian iff this topology is Noetherian. This completes the analogy between equational Noetherian and equational Artinian algebras.
Keywords: algebraic sets, radical ideals, coordinate algebras, Zariski topology, equationally Noetherian algebras, equational Artinian algebras; radical topology.
Mots-clés : algebraic structures, equations 
@article{SEMR_2016_13_a23,
     author = {P. Modabberi and M. Shahryari},
     title = {On the equational {Artinian} algebras},
     journal = {Sibirskie \`elektronnye matemati\v{c}eskie izvesti\^a},
     pages = {875--881},
     publisher = {mathdoc},
     volume = {13},
     year = {2016},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/SEMR_2016_13_a23/}
}
TY  - JOUR
AU  - P. Modabberi
AU  - M. Shahryari
TI  - On the equational Artinian algebras
JO  - Sibirskie èlektronnye matematičeskie izvestiâ
PY  - 2016
SP  - 875
EP  - 881
VL  - 13
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/SEMR_2016_13_a23/
LA  - en
ID  - SEMR_2016_13_a23
ER  - 
%0 Journal Article
%A P. Modabberi
%A M. Shahryari
%T On the equational Artinian algebras
%J Sibirskie èlektronnye matematičeskie izvestiâ
%D 2016
%P 875-881
%V 13
%I mathdoc
%U http://geodesic.mathdoc.fr/item/SEMR_2016_13_a23/
%G en
%F SEMR_2016_13_a23
P. Modabberi; M. Shahryari. On the equational Artinian algebras. Sibirskie èlektronnye matematičeskie izvestiâ, Tome 13 (2016), pp. 875-881. http://geodesic.mathdoc.fr/item/SEMR_2016_13_a23/

[1] Baumslag G., Myasnikov A., Remeslennikov V., “Algebraic geometry over groups. I: Algebraic sets and ideal theory”, J. Algebra, 219 (1999), 16–79 | DOI | MR | Zbl

[2] Baumslag G., Myasnikov A., Remeslennikov V., “Discriminating and co-discriminating groups”, J. Group Theory, 3:4 (2000), 467–479 | DOI | MR | Zbl

[3] Baumslag G., Myasnikov A., Romankov V., “Two theorems about equationally Noetherian groups”, J. Algebra, 194 (1997), 654–664 | DOI | MR | Zbl

[4] Burris S., Sankappanavar H. P., A course in universal algebra, Springer-Verlag, 1981 | MR | Zbl

[5] Daniyarova E., Myasnikov A., Remeslennikov V., “Unification theorems in algebraic geometry”, Algebra and Discrete Mathamatics, 1 (2008), 80–112 | MR | Zbl

[6] Daniyarova E., Myasnikov A., Remeslennikov V., “Algebraic geometry over algebraic structures. II: Fundations”, J. Math. Sci., 185:3 (2012), 389–416 | DOI | MR | Zbl

[7] Daniyarova E., Myasnikov A., Remeslennikov V., “Algebraic geometry over algebraic structures. III: Equationally Noetherian property and compactness”, South. Asian Bull. Math., 35:1 (2011), 35–68 | MR | Zbl

[8] Daniyarova E., Myasnikov A., Remeslennikov V., “Algebraic geometry over algebraic structures. IV: Equatinal domains and co-domains”, Algebra and Logic, 49:6 (2010), 483–508 | DOI | MR | Zbl

[9] Daniyarova E., Remeslennikov V., “Bounded algebraic geometry over free Lie algebras”, Algebra and Logic, 44:3 (2005), 148–167 | DOI | MR | Zbl

[10] Kharlampovich O., Myasnikov A., “Tarski's problem about the elementary theory of free groups has a psitive solution”, E.R.A. of AMS, 4 (1998), 101–108 | MR | Zbl

[11] Kharlampovich O., Myasnikov A., “The elemntary theory of free non-abelian groups”, J. Algebra, 302 (2006), 451–552 | DOI | MR | Zbl

[12] Kharlampovich O., Myasnikov A., “Irreducible affine varieties over a free group. I: Irreducibility of quadratic equations and Nullstellensatz”, J. Algebra, 200:2 (1998), 472–516 | DOI | MR | Zbl

[13] Modabberi P., Shahryari M., “Compactness conditions in universal algebraic geometry”, Algebra and Logic, 55:2 (2016), 146–172 | DOI

[14] Myasnikov A., Remeslennikov V., “Algebraic geometry over groups. II: Logical Fundations”, J. Algebra, 234 (2000), 225–276 | DOI | MR | Zbl

[15] Sela Z., Diophantine geometry over groups: I–X, preprints, Arxiv