On strongly coseparable modules

A module M is called $$\mathfrak {s}$$ s -coseparable if for every nonzero submodule U of M such that M/U is finitely generated, there exists a nonzero direct summand V of M such that $$V \subseteq U$$ V ⊆ U and M/V is finitely generated. It is shown that every non-finitely generated free module is $$\mathfrak {s}$$ s -coseparable but a finitely generated free module is not, in general, $$\mathfrak {s}$$ s -coseparable. We prove that the class of $$\mathfrak {s}$$ s -coseparable modules over a right noetherian ring is closed under finite direct sums. We show that the class of commutative rings R for which every cyclic R-module is $$\mathfrak {s}$$ s -coseparable is exactly that of von Neumann regular rings. Some examples of modules M for which every direct summand of M is $$\mathfrak {s}$$ s -coseparable are provided.

Spectrum of the zero-divisor graph of von Neumann regular rings

The zero-divisor graph [Formula: see text] of a commutative ring [Formula: see text] is the graph whose vertices are the nonzero zero divisors in [Formula: see text] and two vertices [Formula: see text] and [Formula: see text] are adjacent if and only if [Formula: see text]. We study the adjacency and Laplacian eigenvalues of the zero-divisor graph [Formula: see text] of a finite commutative von Neumann regular ring [Formula: see text]. We prove that [Formula: see text] is a generalized join of its induced subgraphs. Among the [Formula: see text] eigenvalues (respectively, Laplacian eigenvalues) of [Formula: see text], exactly [Formula: see text] are the eigenvalues of a matrix obtained from the adjacency (respectively, Laplacian) matrix of [Formula: see text]-the zero-divisor graph of nontrivial idempotents in [Formula: see text]. We also determine the degree of each vertex in [Formula: see text], hence the number of edges.

Commutative von Neumann regular rings are 1-Gröbner

Let R be a commutative von Neumann regular ring. We show that every finitely generated ideal I in the ring of polynomials R[X] has a strong Gr?bner basis. We prove this result using only the defining property of a von Neumann regular ring.

From noncommutative diagrams to anti-elementary classes

Anti-elementarity is a strong way of ensuring that a class of structures, in a given first-order language, is not closed under elementary equivalence with respect to any infinitary language of the form [Formula: see text]. We prove that many naturally defined classes are anti-elementary, including the following: the class of all lattices of finitely generated convex [Formula: see text]-subgroups of members of any class of [Formula: see text]-groups containing all Archimedean [Formula: see text]-groups; the class of all semilattices of finitely generated [Formula: see text]-ideals of members of any nontrivial quasivariety of [Formula: see text]-groups; the class of all Stone duals of spectra of MV-algebras — this yields a negative solution to the MV-spectrum Problem; the class of all semilattices of finitely generated two-sided ideals of rings; the class of all semilattices of finitely generated submodules of modules; the class of all monoids encoding the nonstable K0-theory of von Neumann regular rings, respectively, C*-algebras of real rank zero; (assuming arbitrarily large Erdős cardinals) the class of all coordinatizable sectionally complemented modular lattices with a large [Formula: see text]-frame. The main underlying principle is that under quite general conditions, for a functor [Formula: see text], if there exists a noncommutative diagram [Formula: see text] of [Formula: see text], indexed by a common sort of poset called an almost join-semilattice, such that [Formula: see text] is a commutative diagram for every set [Formula: see text], [Formula: see text] for any commutative diagram [Formula: see text] in [Formula: see text], then the range of [Formula: see text] is anti-elementary.