# Ideal property is upper join-closed for Lie rings

This article gives the statement, and possibly proof, of a Lie subring property (i.e., ideal of a Lie ring) satisfying a Lie subring metaproperty (i.e., upper join-closed Lie subring property)
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## Statement

Suppose $L$ is a Lie ring and $I$ is a subring of $L$. Suppose $A_j$ are subrings of $L$ both containing $I$. Then, if $I$ is an ideal of each of the $A_j$s, $I$ is also an ideal of the Lie subring of $L$ generated by the $A_j$s.

## Related facts

### Analogues in other algebraic structures

Similar facts in other algebraic structures:

Opposites in other algebraic structures: