Wedderburn–Malcev decomposition of one-sided ideals of finite dimensional algebras

Abstract

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.

Inner Ideals of The Symplectic Simple Lie Algebra

“…Lemma (2.27) [17] Let be a simple with involution and let be an idempotent in with . Then (1) is a -Lie of both and . ( 2) is a -regular I-ideal of…”

“…Fernández López, García and Gómez Lozano in [8] showed that by using inner ideals we can development structure for Lie algebras to be Artinian. They are also necessary for the development representation theory of nonsemisimple Lie algebras (see [1] for details). In Classical Lie rings, Benkart and Fernández López classified inner ideals (see [4] and [6]), These algebras can be constructed as inferred Lie sub-algebras of simple Artinian rings.…”

Local Systems of Simple Locally Finite Associative Algebras

“…If A is finite dimensional, then it follows by Wedderburn-Malcev Theorem (see [8,Theorem 1]) that there exists a semisimple subalgebra S of A such that A = S ⊕ rad A and for any semisimple subalgebra Q of A, there is r ∈ rad A with Q ⊆ (1 + r)S(1 + r). Recall that the rank of a perfect finite dimensional algebra A is the smallest rank of the simple components of A/ rad A. Theorem 4.5.…”

“…[8, Theorem 6] Let A be a finite dimensional algebra and let I be a left ideal of A. Suppose that A/R is separable.…”

Jordan-Lie inner ideals of finite dimensional associative algebras

“…Then L is said to be X-minimal ifL = X and for every left ideal L ′ of A with L ′ ⊆ L andL ′ = X one has L = L ′ . We will need the following theorem from [3]. (ii) This follows from (i) and Lemma 6.18.…”