Strong Gröbner bases and linear algebra in multivariate polynomial rings over Euclidean domains

We provide a self-contained introduction to Gröbner bases of submodules of $R[x_1,\dots ,x_n{]}^k$, where $R$ is a Euclidean domain, and explain how to use these bases to solve linear systems over $R[x_1,\dots ,x_n]$.