# Dhara-Rehman-Raza's identities on left ideals of prime rings

## TL;DR

This paper demonstrates that square closed Lie ideals in 2-torsion free prime rings necessarily contain nonzero ideals, extending previous results on Jordan ideals to a broader class of identities in ring theory.

## Contribution

It generalizes existing results by showing that square closed Lie ideals in prime rings contain nonzero ideals, broadening the understanding of identities over left ideals.

## Key findings

- Square closed Lie ideals in prime rings contain nonzero ideals.
- Extension of previous results on Jordan ideals to Lie ideals.
- Emphasizes the significance of studying identities over one-sided ideals.

## Abstract

It is known that every nonzero Jordan ideal of $2$-torsion free semiprime rings contains a nonzero ideal. In this paper we show that also any square closed Lie ideal of a $2$-torsion free prime ring contains a nonzero ideal. This can be interpreted by saying that studying identities over one sided ideals is the "optimal" case to study identities. With this fact in mind, we generalize some results of Dhara, Rehman and Raza in [Lie ideals and action of generalized derivations in rings, Miskolc Mathematical Notes, \textbf{16} (2015), 769 -- 779] to the context of nonzero left ideals.