ICM 1998, Berlin, Aug. 18–27

Abstracts of Short Communications and Posters

Section:

2. Algebra

1991 MS Classification: 16Y Kamal, Ahmed A. M., Cairo University, Faculty of Science, Department of Mathematics, Giza, Egypt Strongly right bounded left near-rings The aim of this paper is to introduce the concept of strongly right bounded (SRB) to the variety of left near-rings (l. n. r.) and study some recent results concerning SRB rings for l. n. r. We proved:

– A l. n. r. R is SRB (SRB^{∗}) if and only if every nonzero right ideal (right R-subgroup) of R is an essential extension of a nonzero ideal of R, moreover we characterized the largest ideal of R contained in xR for each nonzero element x in the SRB^{∗} l. n. r. R.

– A l. n. r. R is SRB^{∗} if and only if the prime radical P (R) of R is SRB^{∗} ideal of R and R contains a reduced SRB^{∗} ideal K of R such that P (R)+K is an essential in R as a right R-subgroup of R.

– Every semiprime l. n. r. contains a unique maximal SRB^{∗} ideal.

– Direct sums and direct products of a family of l. n. r. are SRB (SRB^{∗}) if and only if each l. n. r. of the family is SRB (SRB^{∗}).

– A direct summand ideal of an SRB l. n. r. is again an SRB l. n. r.

– The right socle of an SRB l. n. r. is a direct sum of minimal ideals.