Computing Minimal Free Resolutions over Monomial Semirings with Coefficients in D-A Rings

The study of Gröbner–Shirshov bases in a field, where the set of monomials is a semiring, was first extended to valuation rings by Yatma et al. In a subsequent development, S.Diop et al. further generalized these methods to the setting of D-A rings (divisible and annihilable rings), preserving the semiring structure for the set of monomials, whether commutative or not. Using the approach introduced by Diop et al. in the commutative case, we propose a new technique for computing a minimal free resolution for the ideal I as an R module. This extension of the method shows the variety and applicability of the Gröbner–Shirshov basis framework in various algebraic settings.