Theorem. Das Krull-Schmidt Theorem gilt nicht in der Kategorie der Chow Motive M(PGL 1 (A), Z), wobei A eine zentral einfache Algebra vom Grad 5 ist.

We prove that the Krull dimension of the ring of holomorphic functions of a connected complex manifold is at least the cardinality of continuum if and only if it is > 0. Let R be a commutative ring. Recall that the Krull dimension dim(R) of R is the supremum of cardinalities lengths of chains of distinct proper prime ideals in R. Our main result is: Theorem 1. Let M be a connected complex manif...

Abstract We study direct-sum decompositions of torsion-free, finitely generated modules over a (commutative) Bass ring R through the factorization theory corresponding monoid T ( ). Results Levy–Wiegand and Levy–Odenthal together with local case yield an explicit description The is typically neither factorial nor cancellative. Nevertheless, we construct transfer homomorphism to graph agglomerat...

Abstract Let H be a cancellative commutative monoid, let $$\mathcal {A}(H)$$ A ( H ) the set of atoms and $$\widetilde{H}$$ ~ root closure . Then is called transfer Krull if there exists homomor...

We introduce and study in detail the notion of compatibility between valuations orderings real hyperfields. investigate their relation with induced on factor residue Much theory from fields can be generalized to hyperfields; we point out facts that cannot. generalize Baer-Krull theorem

Introduction and summary. In the theory of non-commutative rings certain distinguished subrings, one-sided and two-sided ideals, play the important roles. Ideals combine under crosscut, union and multiplication and hence are an instance of a lattice over which a non-commutative multiplication is defined.f The investigation of such lattices was begun by W. Krull (Krull [3]) who discussed decompo...

The theory of valuations was started in 1912 by the Hungar-ian mathematician Josef K urschh ak who formulated the valuation axioms as we are used today. The main motivation was to provide a solid foundation for the theory of p-adic elds as deened by Kurt Hensel. In the following decades we can observe a quick development of valuation theory , triggered mainly by the discovery that much of algeb...