ISJ Theoretical & Applied Science 07(51) 2017

ISPC Materials and technologies, Philadelphia, USA

Mushrub VA, Sukhorukova IV, Mochalina EP, Ivankova GV

SOME PROPERTIES OF THE LATTICE OF F-CLOSED RIGHT IDEALS.

Full Article: PDF

Scientific Object Identifier: http://s-o-i.org/1.1/TAS-07-51-17

DOI: https://dx.doi.org/10.15863/TAS.2017.07.51.17

Language: English

Citation: Mushrub VA, Sukhorukova IV, Mochalina EP, Ivankova GV (2017) SOME PROPERTIES OF THE LATTICE OF F-CLOSED RIGHT IDEALS. ISJ Theoretical & Applied Science, 07 (51): 103-106. Soi: http://s-o-i.org/1.1/TAS-07-51-17 Doi: https://dx.doi.org/10.15863/TAS.2017.07.51.17

Pages: 103-106

Published: 30.07.2017

Abstract: Throughout this paper R is a unitary associative ring and f is an injective ring endomorphiosm of R. In the present article, we introduce the notion of the lattice Lat(R, f ) of all f-closed right ideals of R with some special operation instead of the intersection operation. The paper is devoted to the study of this lattice. In particular, we investigate the interrelationship between the lattice of all f-closed right ideals of R and the lattice of right ideals of the Cohn-Jordan extension A. We obtained some results in this direction. In Theorem 1 we give necessary and sufficient conditions, in terms of the lattice Lat(R, f ), for the Cohn-Jordan extension A be a right Artinian ring. This theorem implies in particular that A is right Artinian provided that R is right Artinian. Theorem 2 is a structural theorem and states that a ring R with a bounded length of chains of the right f-closed ideals is embeddable in a semisimple Artinian ring. The authors’ original proof is based on the Cohn-Jordan extension. The Cohn-Jordan extensions were first introduced in [8] for the study of skew polynomial rings constructed by means of a ring endomorphism. Five open questions are formulated.

Key words: lattice, composition length, right Artinian rings