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Let $G$ be a direct product of inner forms of general linear groups overnon-archimedean locally compact fields of residue characteristic $p$ and let$K^1$ be the pro-$p$-radical of a maximal compact open subgroup of $G$. In thispaper we describe the (intertwining) Hecke algebra $\mathscr{H}(G,K^1)$, thatis the convolution $\mathbb{Z}$-algebra of functions from $G$ to $\mathbb{Z}$that are bi-invariant for $K^1$ and whose supports are a finite union of$K^1$-double cosets. We produce a presentation by generators and relations ofthis algebra. Finally we prove that the level-$0$ subcategory of the categoryof smooth representations of $G$ over a unitary commutative ring $R$ such that$p\in R^{\times}$ is equivalent to the category of modules over$\mathscr{H}(G,K^1)\otimes_\mathbb{Z} R$.

Hecke algebra with respect to the pro-$p$-radical of a maximal compact open subgroup for $GL(n,F)$ and its inner forms