Abstract
The Kurosch-Ore theorem1 asserts that if an element of a modular lattice has two decompositions into irreducibles, then each irreducible of one decomposition may be replaced by a suitably chosen irreducible from the other decomposition. It follows that the number of irreducibles in the two decompositions is the same.
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References
G. Birkhoff, Lattice theory, Amer. Math. Soc. Colloquium Publications, vol. 25, 1940.
R.P. Dilworth, Arithmetical theory of Birkhoff lattices, Duke Math. J. vol. 8 (1941) pp. 286–299.
—, Ideals in Birkhoff lattices, Trans. Amer. Math. Soc. vol. 49 (1941) pp. 325–353.
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Dilworth, R.P. (1990). Note on the Kurosch-Ore Theorem. 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_14
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DOI: https://doi.org/10.1007/978-1-4899-3558-8_14
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