Distributivity, binary relations, and standard bases

In the author's previous papers, the connection between generating syzygy modules by binary relations, the property of a commutative ring to be arithmetical (that is to have a distributive ideal lattice), and the use of the so-called S-polynomials in the standard basis theory were discussed. In this note, these connections are considered in a more general context. As an illustration of the usefulness of these considerations, a simple proof of some well-known fact from commutative algebra is given.