# Generators of existence varieties of regular rings and complemented Arguesian lattices

Christian Herrmann; Marina Semenova

Open Mathematics (2010)

- Volume: 8, Issue: 5, page 827-839
- ISSN: 2391-5455

## Access Full Article

top## Abstract

top## How to cite

topChristian Herrmann, and Marina Semenova. "Generators of existence varieties of regular rings and complemented Arguesian lattices." Open Mathematics 8.5 (2010): 827-839. <http://eudml.org/doc/268988>.

@article{ChristianHerrmann2010,

abstract = {We proved in an earlier work that any existence variety of regular algebras is generated by its simple unital Artinian members, while any existence variety of Arguesian sectionally complemented lattices is generated by its simple members of finite length. A characterization of the class of simple unital Artinian members [members of finite length, respectively] of such varieties is given in the present paper.},

author = {Christian Herrmann, Marina Semenova},

journal = {Open Mathematics},

keywords = {Complemented modular lattice; Regular ring; Existence variety; Matrix ring; complemented modular lattice; von Neumann regular ring; existence variety; matrix ring; complemented Arguesian lattice; continuous geometry; coordinatization of lattice geometries},

language = {eng},

number = {5},

pages = {827-839},

title = {Generators of existence varieties of regular rings and complemented Arguesian lattices},

url = {http://eudml.org/doc/268988},

volume = {8},

year = {2010},

}

TY - JOUR

AU - Christian Herrmann

AU - Marina Semenova

TI - Generators of existence varieties of regular rings and complemented Arguesian lattices

JO - Open Mathematics

PY - 2010

VL - 8

IS - 5

SP - 827

EP - 839

AB - We proved in an earlier work that any existence variety of regular algebras is generated by its simple unital Artinian members, while any existence variety of Arguesian sectionally complemented lattices is generated by its simple members of finite length. A characterization of the class of simple unital Artinian members [members of finite length, respectively] of such varieties is given in the present paper.

LA - eng

KW - Complemented modular lattice; Regular ring; Existence variety; Matrix ring; complemented modular lattice; von Neumann regular ring; existence variety; matrix ring; complemented Arguesian lattice; continuous geometry; coordinatization of lattice geometries

UR - http://eudml.org/doc/268988

ER -

## References

top- [1] Birkhoff G., Lattice Theory, 3rd ed., American Mathematical Society Colloquium Publications, 25, American Mathematical Society, Providence, 1967
- [2] Cohn P.M., Introduction to Ring Theory, Springer Undergraduate Mathematics Series, Springer, London, 2000 Zbl0937.16001
- [3] Crawley P., Dilworth R.P., Algebraic Theory of Lattices, Prentice-Hall, Englewood Cliffs, 1973 Zbl0494.06001
- [4] Goodearl K.R., Von Neumann Regular Rings, 2nd ed., Robert E. Krieger, Malabar, 1991
- [5] Goodearl K.R., Menal P., Moncasi J., Free and residually Artinian regular rings, J. Algebra, 1993, 156(2), 407–432 http://dx.doi.org/10.1006/jabr.1993.1082 Zbl0780.16006
- [6] Hall T.E., Identities for existence varieties of regular semigroups, Bull. Austral. Math. Soc., 1989, 40(1), 59–77 http://dx.doi.org/10.1017/S000497270000349X Zbl0666.20028
- [7] Herrmann C., Generators for complemented modular lattices and the von Neumann-Jónsson coordinatization theorems, Algebra Universalis, (in press), DOI: 10.1007/s00012-010-0064-5 Zbl1205.06004
- [8] Herrmann C., Huhn A.P., Zum Begriff der Charakteristik modularer Verbände, Math. Z., 1975, 144(3), 185–194 http://dx.doi.org/10.1007/BF01214134 Zbl0316.06006
- [9] Herrmann C., Semenova M., Existence varieties of regular rings and complemented modular lattices, J. Algebra, 2007, 314(1), 235–251 http://dx.doi.org/10.1016/j.jalgebra.2007.01.038 Zbl1139.06003
- [10] Jipsen P., Rose H., Varieties of Lattices, Lecture Notes in Mathematics, 1533, Springer, Berlin, 1992 Zbl0779.06005
- [11] Jónsson B., Representations of complemented modular lattices, Trans. Amer. Math. Soc., 1960, 97, 64–94 http://dx.doi.org/10.2307/1993364 Zbl0101.02204
- [12] Kad’ourek J., Szendrei M.B., On existence varieties of E-solid semigroups, Semigroup Forum, 1999, 59(3), 470–521 http://dx.doi.org/10.1007/s002339900066
- [13] Maeda F., Kontinuierliche Geometrien, Die Grundlehren der mathematischen Wissenschaften, 95, Springer, Berlin, 1958
- [14] Mal’cev A.I., Algebraic Systems, Die Grundlehren der mathematischen Wissenschaften, 192, Springer, Berlin, 1973
- [15] von Neumann J., Continuous Geometry, Princeton Mathematical Series, 25, Princeton University Press, Princeton, 1960
- [16] Skornyakov L.A., Complemented Modular Lattices and Regular Rings, Oliver & Boyd, Edinburgh-London, 1964 Zbl0156.04101
- [17] Veblen O., Young J.W., Projective Geometry I, Ginn & Co., Boston, 1910

## NotesEmbed ?

topTo embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.