Maximal zero product subrings and inner ideals of simple rings

Abstract

Let Q be a (non-unital) simple ring. A nonempty subset S of Q is said to have zero product if S 2 = 0. We classify all maximal zero product subsets of Q by proving that the map R → R ∩ LeftAnn(R) is a bijection from the set of all proper nonzero annihilator right ideals of Q onto the set of all maximal zero product subsets of Q. We also describe the relationship between the maximal zero product subsets of Q and the maximal inner ideals of its associated Lie algebra.

Jordan-Lie inner ideals of finite dimensional associative algebras

“…Lemmas 4.6 and 4.7 imply that all Jordan-Lie inner ideals of A (−) are generated by idempotents, which is essentially known, see for example [12, Theorem 6.1 (2)]. We summarize description of Jordan-Lie inner ideals of A (k) in the following proposition.…”

“…In this section we describe bar-minimal regular inner ideals of A (−) and A (1) . We start with the following result which is a slight generalization of [ Regular inner ideals were first defined in [4] (in characteristic zero) and were recently used in [2] to classify maximal zero product subsets of simple rings. Note that every regular inner ideal is Jordan-Lie (see Lemma 6.15).…”