If A is a unital associative ring and ℓ ≥ 2, then the general linear group GL(ℓ, A) has root subgroups Uα and Weyl elements nα for α from the root system of type Aℓ−1. Conversely, if an arbitrary group has such root subgroups and Weyl elements for ℓ ≥ 4 satisfying natural conditions, then there is a way to recover the ring A. A generalization of this result not involving the Weyl elements is proved, so instead of the matrix ring M(ℓ, A), a nonunital associative ring with a well-behaved Peirce decomposition is provided. © 2024 American Mathematical Society