For every number n≥2, let Γn be the hypergraph on Rn of arity four consisting all non-degenerate Euclidean rectangles. It is consistent with ZF+DC set theory that chromatic countable while Γn+1 not.

Let R be a commutative local k-algebra of Krull dimension one, where k is field. α automorphism R, and define S to the skew polynomial algebra R[θ;α]. We offer, under some additional assumptions on criterion for have injective hulls all simple S-modules locally Artinian - that is, satisfy property (⋄). It easy well known if finite order, then has this property, but in order get when infinite we...

Let D be a Krull domain and Int(D) the ring of integer-valued polynomials on D. For any f ∈ Int(D), we explicitly construct a divisor homomorphism from [[f ]], the divisor-closed submonoid of Int(D) generated by f , to a finite sum of copies of (N0,+). This implies that [[f ]] is a Krull monoid. For V a discrete valuation domain, we give explicit divisor theories of various submonoids of Int(V ...

A monoid M is a Cale monoid with base Q if for every nonunit x ∈ M there exists a positive integer n such that xn factors uniquely up to order and associates as elements from Q ⊆ M\M. An integral domain D is a Cale domain with base Q, if its multiplicative monoid of nonzero elements is a Cale monoid with base Q. We explore the basic properties of Cale monoids and integral domains. In particular...

The principle “Every result in classical homological algebra should have a counterpart in Gorenstein homological algebra” was given by Henrik Holm. There is a remarkable body of evidence supporting this claim. Perhaps one of the most glaring exceptions is provided by the fact that tensor products of Gorenstein projective modules need not be Gorenstein projective, even over Gorenstein rings. So ...

In this paper we study a representation of a fuzzy subgroup $mu$ of a group $G$, as a product of indecomposable fuzzy subgroups called the components of $mu$.  This representation is unique up to the number of components and their isomorphic copies. In the crisp group theory, this is a well-known Theorem attributed to Remak, Krull, and Schmidt. We consider the lattice of fuzzy subgroups and som...