Where can I find the proof of Kurosch-Ore theorem in lattice theory? The statement of this theorem is: Let $L$ be a modular lattice with $0$ and $1$ that satisfies both chain conditions. Then for any element $a$ in the lattice, any two decompsiton of $a$ into independent and indecomposable element can be put in 1-1 correspondence.
2 Answers
You might prefer the treatment in Chapter 2, section 2.3, of Algebras, Lattices, and Varieties by McKenzie, McNulty, and Taylor. They develop the Kurosh-Ore theorem along with Dedekind's Transposition Principle toward a more general result on modular lattices (with finite chain condition), namely the Direct Join Decomposition Theorem, where Jonsson's proof is used to inspire the presentation. DJDT takes up a lot of section 2.4, and I recommend two or more readings to appreciate it.
Gerhard "Needs Five Or More Readings" Paseman, 2015.12.29
Birkhoff's Lattice Theory has a proof, summarized as follows by C. Faith, Algebra II Ring Theory:
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$\begingroup$ But here we need each component are independent, that is, just like krull-schmidt decomposition in group theory, we need the intersection of any two components is 0. In Jacobson's book (GTM 30, p204). He has mentioned this theorem without proof. $\endgroup$ Dec 29, 2015 at 11:19