Among reduced Noetherian prime characteristic commutative rings, we prove that a regular ring is precisely that where the finite intersection of ideals commutes with taking bracket powers. However, reducedness is essential for this equivalence. Connections are made with Ohm-Rush content theory, intersection-flatness of the Frobenius map, and various flatness criteria.
Let be a complete multipartite graph on with and being its binomial edge ideal. It is proved that the Castelnuovo-Mumford regularity is for any positive integer .
A weak basis of a module is a generating set of the module minimal with respect to inclusion. A module is said to be regularly weakly based provided that each of its generating sets contains a weak basis. We study (1) rings over which all modules are regularly weakly based, refining results of Nashier and Nichols, and (2) regularly weakly based modules over Dedekind domains.
For any positive power n of a prime p we find a complete set of generating relations between the elements [r] = rⁿ - r and p·1 of a unitary commutative ring.
We prove that generating relations between the elements [r] = r²-r of a commutative ring are the following: [r+s] = [r]+[s]+rs[2] and [rs] = r²[s]+s[r].
We study finitely generated bigraded Buchsbaum modules over a standard bigraded polynomial ring with respect to one of the irrelevant bigraded ideals. The regularity and the Hilbert function of graded components of local cohomology at the finiteness dimension level are considered.
Let be a commutative Noetherian ring and let be a semidualizing -module. We prove a result about the covering properties of the class of relative Gorenstein injective modules with respect to which is a generalization of Theorem 1 by Enochs and Iacob (2015). Specifically, we prove that if for every -injective module , the character module is -flat, then the class is closed under direct sums and direct limits. Also, it is proved that under the above hypotheses the class is covering....
We study the construction of new multiplication modules relative to a torsion theory . As a consequence, -finitely generated modules over a Dedekind domain are completely determined. We relate the relative multiplication modules to the distributive ones.
Let be a commutative Noetherian ring, and let be a semidualizing -module. The notion of -tilting -modules is introduced as the relative setting of the notion of tilting -modules with respect to . Some properties of tilting and -tilting modules and the relations between them are mentioned. It is shown that every finitely generated -tilting -module is -projective. Finally, we investigate some kernel subcategories related to -tilting modules.
We prove a convenient equivalent criterion for monotone completeness of ordered fields of generalized power series with exponents in a totally ordered Abelian group G and coefficients in an ordered field F. This enables us to provide examples of such fields (monotone complete or otherwise) with or without integer parts, i.e. discrete subrings approximating each element within 1. We include a new and more straightforward proof that is always Scott complete. In contrast, the Puiseux series field...