Hostname: page-component-cd9895bd7-mkpzs Total loading time: 0 Render date: 2024-12-25T08:48:25.844Z Has data issue: false hasContentIssue false
  • English
  • Français

Minimal ideals in near-rings and localized distributivity conditions

Published online by Cambridge University Press:  09 April 2009

Gary Birkenmeier  and
Henry Heatherly
Gary Birkenmeier
Affiliation:
University of Southwestern Louisiana Lafayette, Louisiana 70504, U.S.A.
Henry Heatherly
Affiliation:
University of Southwestern Louisiana Lafayette, Louisiana 70504, U.S.A.
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

Let B, S, and T be subsets of a (left) near-ring R with B and T nonempty. We say B is (S, T)-distributive if s(b1+b2)t = sb1t + sb2t, for each s ∈ S, b1, 2 ∈ B, t ∈ T. Basic properties for this type of ‘localized distributivity’ condition are developed, examples are given, and applications are made in determining the structure of minimal ideals. Theorem. If I is a minimal ideal of R and Ik is (Im, In)-distributive for some k, n ≧ 1, m ≧ 0, then either I2 = 0 or I is a simple, nonnilpotent ring with every element of I distributive in R. Theorem. Let Rk be (Rm, Rn)-distributive, for some k, n ≧ 1, m ≧ 0; if R is semiprime or is a subdirect product of simple near-rings, then R is a ring. Connections are established with near-rings which satisfy a permutation identity and with weakly distributive near-rings. If R → A → 0 is an exact sequence of near-rings, then conditions on A are given which will impose conditions on the minimal ideals of R.

Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 54 , Issue 2 , April 1993 , pp. 156 - 168
Copyright
Copyright © Australian Mathematical Society 1993

References

[1]Andrunakievic, V., ‘Radicals in associative rings I’, Mat. Sb. 44 (1958), 179–212.Google Scholar
[2]Birkenmeier, G. and Heatherly, H., ‘Polynomial identity properties for near-rings on certain groups’, Near-Rings Newsletter 12 (1989), 5–15.Google Scholar
[3]Birkenmeier, G. and Heatherly, H., ‘Medical near-rings’, Monatsh. Math. 107 (1989), 89–110.CrossRefGoogle Scholar
[4]Birkenmeier, G. and Heatherly, H., ‘Medial and distributively generated near-rings’, Monatsh. Math. 109 (1990), 97–101.Google Scholar
[5]Birkenmeier, G. and Heatherly, H., ‘Medial rings and an associated radical’, Czechoslovak Math. J. 40 (1990), 258–283.CrossRefGoogle Scholar
[6]Birkenmeier, G. and Heatherly, H., ‘Permutation identity rings and the medial radical’, in: Proc. Conference on Non-Commutative Ring Theory (Athens, Ohio), Lecture Notes in Math. 1448 (Springer, Berlin, 1990), pp. 125–138.CrossRefGoogle Scholar
[7]Birkenmeier, G. and Heatherly, H., ‘Minimal ideals in near-rings’, Comm. Algebra 20 (1992), 457–468.CrossRefGoogle Scholar
[8]Brown, H., ‘Near algebras’, Illinois J. Math. 12 (1968), 215–227.CrossRefGoogle Scholar
[9]Brown, H., ‘Distributor theory in near-rings’, Comm. Pure Math. 21 (1968), 535–544.Google Scholar
[10]Clay, J. R., ‘Near-rings on groups of low order’, Math. Z. 104 (1968), 364–371.CrossRefGoogle Scholar
[11]Divinsky, N., Rings and Radicals (Univ. of Toronto Press, Ontario, 1965).Google Scholar
[12]Fröhlich, A., ‘Distributively generated near-rings (I. Ideal theory)’, Proc. London Math. Soc. 8 (1958), 76–94.CrossRefGoogle Scholar
[13]Heatherly, H. and Ligh, S., ‘Pseudo-distributive near-rings’, Bull. Austral. Math. Soc. 12 (1975), 449–456.CrossRefGoogle Scholar
[14]Kaarli, K., ‘Minimal ideals in near-rings’, Tartu Riikl. Ül. Toimetised, 336 (1975), 105–142 (in Russian).Google Scholar
[15]Kaarli, K., ‘On Jacobson type radicals of near-rings’, Acta Math. Hung. 50 (1987), 71–78.CrossRefGoogle Scholar
[16]Kaarli, K., ‘On minimal ideals of distributively generated near-rings’, preprint.Google Scholar
[17]Meldrum, J. D. P., Near-rings and their links with groups (Pitman, Marshfield, MA, 1985).Google Scholar
[18]Putcha, M. and Yaqub, A., ‘Semigroups satisfying permutation identities’, Semigroup Forum 3 (1971), 68–73.Google Scholar
[19]Scott, S. D., ‘Minimal ideals in near-rings with minimal condition’, J. London Math. Soc. (2) 8 (1974), 8–12.Google Scholar
[20]Wiegandt, R., ‘On subdirectly irreducible near-rings which are fields’, in: Near-Rings and Near-Fields (ed. Betsch, G.) (Amsterdam, North-Holland, 1987) pp. 295–298.Google Scholar