Medial near-rings

Abstract

Abstract. In this paper we discuss (left) near-rings satisfying the identities: show that N(R) is an ideal, and characterize the simple and subdirectly irreducible near-rings. "Natural" examples from analysis and geometry are produced via a general construction method.

Medial and distributively generated near-rings

“…Properties of these additive groups are then used to obtain results about d.g., medial, or subdirectly irreducible near-rings or near-rings whose additive group is locally nilpotent. This extends results in [1]. We also correct some minor errors found in [1].…”

“…Thus S is a left ideal of R and S (9 (R)) ___ S; since S is normal in R + this yields S (ngp (9 (R))) __c S. From Proposition 2.1 (iii) of [1] we have that [R, R] _c S and S a left ideal implies S is an ideal and a right R-subgroup. If R is d.g., the gp (9 (R)) = R and every normal right R-subgroup is a tight ideal.…”

“…We also correct some minor errors found in [1]. The notation and terminology herein are the same as that used in our previous paper on medial near-tings [1].…”

Left self distributive near-rings

“…Fifteen of these are LSD, of which five are Malone trivial. [4] (ii) On C 6 there are sixty non-isomorphic near-rings [9]. Forty-seven of these are LSD, of which twenty-one are Malone trivial.…”

“…Semigroups which are LSD were investigated by Kepka [14] and LSD rings were investigated by the present authors and Kepka [7]. Previously we examined in depth the properties of rings or near-rings, satisfying one of 1.2, 1.3 or 1.4 [4,5,6]. Near-rings which are both LSD and RSD, herein called self distributive, were considered by Ferrero Cotti [12] and Scapellato [25].…”