Where can I find the proof of Kurosh-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 decompositons of $a$ into independent and indecomposable elements can be put in 1-1 correspondence.
In Jacobson's book (GTM30 p204), he has mentioned this theorem without proof.
Refer the book Gabor Szasz - Introduction to Lattice Theory, Academic Press, N.Y., 1963 for proof of Kurosh-Ore theorem and refer the book P. Crawley and R.P, Dilworth - Algebraic Theory of Lattices, Prentice - Hall Inc., N. J., 1973, for proof of the Ore's theorem