A basis for Numerical Functionals
A basis for the ring of doubly integer-valued polynomials.
A Bertini type theorem in analytic geometry.
A Bezout theorem for determinantal modules
A broken circuit ring.
A calculation of injective dimension over valuation domains
A Cancellation Theorem for Artinian Local Algebras.
A Cancellation Theorem for Projective Modules in the Metastable Range.
A Change of Ring Theorem with Applications to Poincaré Series and Intersection Multiplicity.
A characterization of commutative Fréchet algebras with all ideals closed
Let A be a commutative unital Fréchet algebra, i.e. a completely metrizable topological algebra. Our main result states that all ideals in A are closed if and only if A is a noetherian algebra
A characterization of F-algebras with all one-sided ideals closed
We prove that a real or complex F-algebra has all left and right ideals closed if and only if it is noetherian.
A characterization of Krull rings with zero divisors
It is proved that a Marot ring is a Krull ring if and only if its monoid of regular elements is a Krull monoid.
A Characterization of One-Element p-Bases of Rings of Constants
Let K be a unique factorization domain of characteristic p > 0, and let f ∈ K[x₁,...,xₙ] be a polynomial not lying in . We prove that is the ring of constants of a K-derivation of K[x₁,...,xₙ] if and only if all the partial derivatives of f are relatively prime. The proof is based on a generalization of Freudenburg’s lemma to the case of polynomials over a unique factorization domain of arbitrary characteristic.
A characterization of partition polynomials and good Bernoulli trial measures in many symbols
Consider an experiment with d+1 possible outcomes, d of which occur with probabilities . If we consider a large number of independent occurrences of this experiment, the probability of any event in the resulting space is a polynomial in . We characterize those polynomials which arise as the probability of such an event. We use this to characterize those x⃗ for which the measure resulting from an infinite sequence of such trials is good in the sense of Akin.
A characterization of p-bases of rings of constants
We obtain two equivalent conditions for m polynomials in n variables to form a p-basis of a ring of constants of some polynomial K-derivation, where K is a unique factorization domain of characteristic p > 0. One of these conditions involves Jacobians while the other some properties of factors. In the case m = n this extends the known theorem of Nousiainen, and we obtain a new formulation of the Jacobian conjecture in positive characteristic.
A characterization of Prüfer rings.
A class of counterexamples to the Cancellation Problem for arbitrary rings
We present a class of counterexamples to the Cancellation Problem over arbitrary commutative rings, using non-free stably free modules and locally nilpotent derivations.
A class of finite rings
A class of multiplicative lattices
We study the multiplicative lattices which satisfy the condition for all . Call them sharp lattices. We prove that every totally ordered sharp lattice is isomorphic to the ideal lattice of a valuation domain with value group or . A sharp lattice localized at its maximal elements are totally ordered sharp lattices. The converse is true if has finite character.
A class of principal ideal rings arising from the converse of the Chinese remainder theorem.