Derived subring of a Lie ring

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Definition

Let L be a Lie ring. The derived subring or commutator subring of L, denoted L', or [L,L] is defined in the following ways:

  • It is the additive subgroup generated by all elements of the form [x,y], where x,y \in L
  • It is the Lie subring generated by all elements of the form [x,y], where x,y \in L
  • It is the Lie ideal generated by all elements of the form [x,y], where x,y \in L

In situations where the Lie ring is an algebra over some field or ring, the derived subring is also a subalgebra and an ideal over that field or ring. In those cases, it may be termed the derived subalgebra or commutator subalgebra.