Let A be a finite dimensional associative algebra over a perfect field and let R be the radical of A. We show that for every one-sided ideal I of A there exists a semisimple subalgebra S of A such that I = I S ⊕ I R where I S = I ∩ S and I R = I ∩ R.
We study Jordan-Lie inner ideals of finite dimensional associative algebras and the corresponding Lie algebras and show that they admit Levi decompositions. Moreover, we classify Jordan-Lie inner ideals satisfying a certain minimality condition and show that they are generated by pairs of idempotents.
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