The UFD property for group rings was characterized by Gilmer and Parker in 1974. In this paper we give a direct proof of this result by extending it to nonunique factorization setting.
Let A be a discrete valuation domain. It is known that an Eisenstein polynomial f in A[x] is useful in constructing a totally ramified extension B of A, where B similar to= A[x]/(f ) and where the prime of B is the coset x & strns; of x in B. We define a counter-Eisenstein formal power series c E A[x], where A is a complete discrete valuation domain. The definition depends on the choice of a prime p E A and a so-called set of representatives S subset of A. We show that such a counter-Eisenstein formal power series c play a role analogous to that played by an Eisenstein polynomial f in A[x]. Finally, we argue that there is a natural bijective correspondence between the counter-Eisenstein formal power series c's in A[x]and the Eisenstein polynomials f's in A[x] with the property that A[x]/(f) similar to= A[x]/(c) when f corresponds to c.
Let G be an additive abelian group. A sequence S = g(1) & centerdot; ...& centerdot; g(& ell; )of terms from G is a plus-minus weighted zero-sum sequence if there are epsilon(1), ..., epsilon(& ell;) is an element of {-1, 1} such that epsilon(1)g(1) + & centerdot;& centerdot; & centerdot;+ epsilon(& ell;)g(& ell;)= 0. We study sets of lengths in the monoid B +/-(G) of plus-minus weighted zero-sum sequences over G. If G is finite, then sets of lengths are highly structured. If G is infinite, then every finite, nonempty subset of N (>= 2) is the set of lengths of some sequence S is an element of B-+/-(G).
We begin the study of Lipschitz saturation for germs of toric singularities. By looking at their associated analytic algebras, we prove that if (X, 0) is a germ of toric singularity with smooth normalization then its Lipschitz saturation is again toric. Finally we show how to calculate the Lipschitz saturation for some families of toric singularities starting from the semigroup that defines them.
We give a short overview of the contribution to abelian groups, endomorphism rings and modules over commutative rings by the well-known American mathematician Patrick W. Keef on the celebration of his 70th birthday.
