Module over a Lie ring

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Symbol-free definition

A module over a Lie ring is an abelian group along with a homomorphism of Lie rings from the Lie ring to the endomorphism ring of the abelian group, with the Lie bracket given by the additive commutator.

Definition with symbols

Let L be a Lie ring and A be an abelian group. A module structure of A over L is a map:

\cdot:L \times A \to A

satisfying the following:

  1. For every l \in L, the map a \mapsto l \cdot a is an endomorphism of the abelian group A.
  2. For every a \in A, the map l \mapsto l \cdot a is a homomorphism of groups from the additive group of L to A.
  3. For l_1,l_2 \in L and a \in A, we have [l_1,l_2] \cdot a = l_1 \cdot (l_2 \cdot a) - l_2 \cdot (l_1 \cdot a).