Beiträge zur Algebra und Geometrie
Contributions to Algebra and Geometry
38(2), 217-225 (1997)

Minimal and Maximal Ideals in Rings with Involution

Gary F. Birkenmeier, Nico J. Groenewald, Henry E. Heatherly

Mathematics Department, University of Southwestern Louisiana
P.O.Box 41010, Lafayette, La 70504 USA

Department of Mathematics, University of Port Elizabeth
P.O.Box 1600, Port Elizabeth, 6000 South Africa

Abstract: We investigate the relationships between the ideal structure and the $*$-ideal structure of a ring with involution ($*$). Descriptions of $*$-minimal and $*$-maximal ideals are obtained in terms of minimal and maximal ideals, respectively. Furthermore conditions are provided allowing us to associate with each minimal or maximal ideal a $*$-minimal or $*$-maximal ideal, respectively. These connections are strong enough to permit the transfer of various properties from the $*$-ideal structure to the ideal structure or vice-versa.

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