Abstract
In the theory of non-commutative rings certain distinguished subrings, one-sided and two-sided ideals, play the important roles. Ideals combine under crosscut, union and multiplication and hence are an instance of a lattice over which a non-commutative multiplication is defined.† The investigation of such lattices was begun by W. Krull (Krull [3]) who discussed decomposition into isolated component ideals. Our aim in this paper differs from that of Krull in that we shall be particularly interested in the lattice structure of these domains although certain related arithmetical questions are discussed.
Presented to the Society in two parts: April 9, 1938, under the title Non-commutative residuation, and November 26, 1938, under the title Archimedian residuated lattices; received by the editors May 1, 1939.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
G. Birkhoff, Bulletin of the American Mathematical Society, vol. 40 (1934), pp. 613–619.
R.P. Dilworth, Bulletin of the American Mathematical Society, vol. 44 (1938), pp. 262–267.
W. Krull, Mathematische Zeitschrift, vol. 28 (1928), pp. 481–503.
O. Ore. Annals of Mathematics, (2), vol. 36 (1935), pp. 406–432.
M. Ward, Annals of Mathematics, (2), vol. 39 (1938), pp. 558–568.
M. Ward and R.P. Dilworth, Proceedings of the National Academy of Sciences, vol. 24 (1938), pp. 162–164.
R.P. Dilworth —, these Transactions, vol. 45 (1939), pp. 335–354.
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 1990 Springer Science+Business Media New York
About this chapter
Cite this chapter
Dilworth, R.P. (1990). Non-Commutative Residuated Lattices. In: Bogart, K.P., Freese, R., Kung, J.P.S. (eds) The Dilworth Theorems. Contemporary Mathematicians. Birkhäuser, Boston, MA. https://doi.org/10.1007/978-1-4899-3558-8_33
Download citation
DOI: https://doi.org/10.1007/978-1-4899-3558-8_33
Publisher Name: Birkhäuser, Boston, MA
Print ISBN: 978-1-4899-3560-1
Online ISBN: 978-1-4899-3558-8
eBook Packages: Springer Book Archive