Abstract
This paper is a continuation of two previous works studying the units of a compatible nearring R satisfying the descending chain condition on right ideals using a faithful compatible module G of R. A crucial point in doing this involves determining 1 + Ann R (G/H) where H is a direct sum of isomorphic minimal R-ideals. The high point of this paper is extending this determination from the cases in the previous works to the case where G/H and H contain no isomorphic minimal factors. We also shall further expand our knowledge of when a special type of principal series for G introduced in the second of these previous works called a quasi c-chain exists.
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Aichinger, E., Binder, F., Ecker, J., Mayr, P., Nöbauer, C.: SONATA—system of near-rings and their applications, GAP package, Version 2. http://www.algebra.uni-linz.ac.at/Sonata/ (2003)
Aichinger E., Mayr P.: Polynomial functions and endomorphism near-rings on certain linear groups. Comm. Algebra 31, 5627–5651 (2003)
Aichinger E., Mayr P., Meldrum J., Peterson G., Scott S.: Units of compatible nearrings. Monatsh. Math. 164, 119–132 (2011)
Huppert B.: Endliche Gruppen I. Springer, Berlin (1967)
King, M.: The endomorphism near ring on the quaternion group. Master’s Thesis, Texas A&M University (1969)
Lyons C., Malone J.: Finite dihedral groups and d.g. near rings I. Compositio Math. 24, 305–312 (1972)
Meldrum J.: Nearrings and their links with groups. Pitman, Boston (1985)
Peterson G.: On the structure of an endomorphism nearring. Proc. Edinb. Math. Soc. 32, 223–229 (1989)
Peterson G.: Projection idempotents in nearrings. Algebra Colloq. 10, 209–218 (2003)
Peterson G.: Some problems in the theory of nearring modules. Math. Pannon. 20, 109–121 (2006)
Peterson G.: Centralizers and the isomorphism problem for nearrings. Math. Pannon. 17, 3–16 (2006)
Peterson, G., Scott, S.: Units of compatible nearrings, II. Monatsh. Math. (to appear)
Pilz G.: Near-rings, rev. edn. North-Holland, Amsterdam (1983)
Robinson D.: A course in the theory of groups. Springer, Berlin (1993)
Scott S.: Tame near-rings and N-groups. Proc. Edinb. Math. Soc. 23, 275–296 (1980)
Scott S.: N-solubility and N-nilpotency in tame N-groups. Algebra Colloq. 5, 425–448 (1998)
Scott, S.: Some compatible nearring extension theory (preprint)
Scott, S.: Compatibility and centralizer index (preprint)
Scott W.: Group theory. Prentice Hall, Englewood Cliffs (1964)
Thomas A., Wood G.: Group tables. Shiva, Nantwich (1980)
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Communicated by J. S. Wilson.
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Peterson, G.L., Scott, S.D. Units of compatible nearrings, III. Monatsh Math 171, 103–124 (2013). https://doi.org/10.1007/s00605-012-0421-x
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DOI: https://doi.org/10.1007/s00605-012-0421-x