Internal semidirect product of Lie rings

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Definition

Suppose L is a Lie ring and A and B are Lie subrings of L. We say that L is an internal semidirect product of A and B, denoted L = A \rtimes B, if it satisfies all the following conditions:

  1. A is an ideal in L.
  2. B is a subring in L.
  3. A \cap B is the zero subring of L, i.e., it is the set \{ 0 \}.
  4. L = A + B.

Conditions (3) and (4) basically say that the additive group of L is the internal direct product of the additive groups of A and B.

This notion is equivalent to the notion of external semidirect product of Lie rings, via equivalence of internal and external semidirect product for Lie rings.