A zero-dimensional valuation ring is 1-Gröbner

In this study, we consider the following conjecture and open problem: (1) A valuation ring R is 1-Gröbner if and only if R is an Archimedean ring; (2) Is a 1-Gröbner valuation ring coherent? First, we prove that a zero-dimensional valuation ring is 1-Gröbner. Furthermore, we give a positive answer to (1) and a negative answer to (2). Finally, we present an example of a valuation ring that is 1-Gröbner but not coherent or Noetherian.