In this paper, we extend the characterization of Z[x]/〈f〉, where f ∈ Z[x] to be a free Z-module to multivariate polynomial rings over any commutative Noetherian ring, A. The characterization allows us to extend the Gröbner basis method of computing a k-vector space basis of residue class polynomial rings over a field k (Macaulay-Buchberger Basis Theorem) to rings, i.e. A[x1, . . . , xn]/a, wher...

In 1974, the author proved that codimension of ideal (g1,g2,…,gd) generated in group algebra K[Zd] over a field K characteristic 0 by generic Laurent polynomials having same Newton polytope Γ is equal to d!×Volume(Γ). Assuming Newtons simplicial and super-convenient (that is, containing some neighborhood origin), strengthens 1974 result explicitly specifying set Bsh monomials cardinality d!×Vol...

Let R be a polynomial ring over a field of dimension n. Let m be the maximal homogeneous ideal of R. Let I be a complete intersection homogeneous ideal of R minimally generated by polynomials of degrees d1, d2, . . . , dn. F. S. Macaulay showed that for all integers i = 0, 1, . . . , δ + 1, I : m = I + m where δ = d1 + d2 + . . .+ dn − n. We provide a modern proof of a generalization of this re...

We construct explicitly regular sequences in the semigroup ring R = C[K] of lattice points of the graded cone K. We conjecture that the quotients of R by these sequences describe locally string-theoretic cohomology of a toroidal singularity associated to K. As a byproduct, we give an elementary proof of the result of Hochster that semigroup rings of rational polyhedral cones are Cohen-Macaulay.

A result of Watanabe and Yoshida says that an unmixed local ring of positive characteristic is regular if and only if its Hilbert-Kunz multiplicity is one. We show that, for fixed p and d, there exist a number ǫ(d, p) > 0 such that any nonregular unmixed ring R its Hilbert-Kunz multiplicity is at least 1+ ǫ(d, p). We also show that local rings with sufficiently small Hilbert-Kunz multiplicity a...