  The well-known theorem of Shalom--Vaserstein and Ershov--Jaikin-Zapirain
states that the group $\mathrm{EL}_n(\mathcal{R})$, generated by elementary
matrices over a finitely generated commutative ring $\mathcal{R}$, has
Kazhdan's property (T) as soon as $n\geq3$. This is no longer true if the ring
$\mathcal{R}$ is replaced by a commutative rng (a ring but without the
identity) due to nilpotent quotients
$\mathrm{EL}_n(\mathcal{R}/\mathcal{R}^k)$. In this paper, we prove that even
in such a case the group $\mathrm{EL}_n(\mathcal{R})$ satisfies a certain
property that can substitute property (T), provided that $n$ is large enough.
