If
A
A
is a unital associative ring and
ℓ
≥
2
\ell \geq 2
, then the general linear group
GL
(
ℓ
,
A
)
\operatorname {GL}(\ell , A)
has root subgroups
U
α
U_\alpha
and Weyl elements
n
α
n_\alpha
for
α
\alpha
from the root system of type
A
ℓ
−
1
\mathsf A_{\ell - 1}
. Conversely, if an arbitrary group has such root subgroups and Weyl elements for
ℓ
≥
4
\ell \geq 4
satisfying natural conditions, then there is a way to recover the ring
A
A
. A generalization of this result not involving the Weyl elements is proved, so instead of the matrix ring
M
(
ℓ
,
A
)
,
\operatorname {M}(\ell , A),
a nonunital associative ring with a well-behaved Peirce decomposition is provided.