On matrix Lie rings over a commutative ring that contain the special linear Lie ring

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Let $K$ be an associative and commutative ring with $1$, $k$ a subring of $K$ such that $1\in k$, $n\geq 2$ an integer. The paper describes subrings of the general linear Lie ring $gl_{n} ( K )$ that contain the Lie ring of all traceless matrices over $k$.