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.

Higher Braid Groups and Regular Semigroups from Polyadic-Binary Correspondence

In this note, we first consider a ternary matrix group related to the von Neumann regular semigroups and to the Artin braid group (in an algebraic way). The product of a special kind of ternary matrices (idempotent and of finite order) reproduces the regular semigroups and braid groups with their binary multiplication of components. We then generalize the construction to the higher arity case, which allows us to obtain some higher degree versions (in our sense) of the regular semigroups and braid groups. The latter are connected with the generalized polyadic braid equation and R-matrix introduced by the author, which differ from any version of the well-known tetrahedron equation and higher-dimensional analogs of the Yang-Baxter equation, n-simplex equations. The higher degree (in our sense) Coxeter group and symmetry groups are then defined, and it is shown that these are connected only in the non-higher case.

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.