Let S be a polynomial algebra over field. If I is the edge ideal of perfect semiregular tree, then we give precise formulas for values depth, Stanley projective dimension, regularity and Krull dimension S/I.

For a fixed prime p, let H∗(G) denote the mod p cohomology ring of a finite group G, that is, ExtFpG(Fp,Fp) interpreted algebraically and H ∗(K(G, 1); Fp) interpreted topologically. The cohomology ring is a graded commutative Noetherian ring, or equivalently, a finitely generated graded commutative Fp-algebra. Although the ring is not strictly commutative, the usual concepts from commutative ri...

Various topics on affine rings are considered, such as the relationship between Gelfand-Kirillov dimension and Krull dimension, and when a "locally affine" algebra is affine. The dimension result is applied to study prime ideals in fixed rings of finite groups, and in identity components of group-graded rings. 0. Introduction. We study affine rings (i.e., rings finitely generated as algebras ov...

We show that a ring is a Krull ring if and only if every nonzero regular prime ideal contains a t-invertible prime ideal if and only if every proper regular principal ideal is quasi-equal to a product of prime ideals.

First, for a general Krull-Schmidt category, we provide criteria for an Auslander-Reiten component with sections to be standard. Specializing to the category of finitely presented representations of a strongly locally finite quiver and its bounded derived category, we obtain many new types of standard Auslander-Reiten components. Finally, specialized to the module category of a finite-dimension...

Given a set X of “indeterminates” and a field F , an ideal in the polynomial ring R = F [X] is called conservative if it contains with any polynomial all of its monomials. The map S 7→ RS yields an isomorphism between the power setP (X) and the complete lattice of all conservative prime ideals of R. Moreover, the members of any system S ⊆ P (X) of finite character are in one-to-one corresponden...

Let H be a Krull monoid with finite class group G such that every class contains a prime divisor. Then the global tame degree t(H) equals zero if and only if H is factorial (equivalently, |G| = 1). If |G| > 1, then D(G) ≤ t(H) ≤ 1 + D(G) (

We describe prime ideals of height 2 minimally generated by 3 elements in a Gorenstein, Nagata local ring of Krull dimension 3 and multiplicity at most 3. This subject is related to a conjecture of Y. Shimoda and to a long-standing problem of J. Sally.

In this note we show that the quotient field of a domain which is Henselian with respect to a non-trivial ideal is a large field, and give some applications of this fact, using a specialization theorem for ramified covers of the line over (generalized) Krull fields.

Let H be a Krull monoid with infinite class group such that each divisor class contains a prime divisor. We show that for every positive integer n, there exists a divisor closed submonoid S of H such that min∆(S) = n.