  We prove Michael-Simon type Sobolev inequalities for $n$-dimensional
submanifolds in $(n+m)$-dimensional Riemannian manifolds with nonnegative
$k$-th intermediate Ricci curvature by using the Alexandrov-Bakelman-Pucci
method. Here $k=\min(n-1,m-1)$. These inequalities extends Brendle's
Michael-Simon type Sobolev inequalities on Riemannian manifolds with
nonnegative sectional curvature (arXiv:2009.13717) and Dong-Lin-Lu's
Michael-Simon type Sobolev inequalities on Riemannian manifolds with
asymptotically nonnegative sectional curvature (arXiv:2203.14624) to the
$k$-Ricci curvature setting. In particular, a simple application of these
inequalities gives rise to some isoperimetric inequalities for minimal
submanifolds in Riemannian manifolds.
