## Catenarity of monoid algebras

Let R be a commutative ring, let M be a commutative monoid, and let R[M] denote the corresponding monoid algebra. Suppose further that R is universally catenary. One may ask for conditions on M such that the ring R[M] is catenary. I know of the following results: If M is of finite type then R[M] … Read more

## torsion free modules MM over Noetherian domain of dimension 11 for which l(M/aM)≤(dimKK⊗RM)⋅l(R/aR),∀0≠a∈Rl(M/aM) \le (\dim_K K \otimes_R M) \cdot l(R/aR), \forall 0 \ne a \in R

Let R be a Noetherian domain of Krull-dimension 1 (i.e. every non-zero prime ideal maximal). Let M be a torsion free R-module . Let K be the fraction-field of R and let r=dimKS−1M=dimKK⊗RM (where S=R∖{0} ). Suppose r is finite. Under these conditions, if M is finitely generated, then I can prove that l(M/aM)≤r⋅l(R/aR),∀0≠a∈R . … Read more

## Integral domains with totally ordered spectra

In my research I ended up trying to prove some properties of integral domains such that their spectrum is a totally ordered poset. Are there some nice (ubiqitous/natural) examples of such domains, which are not valuation domains? (Of course any local 1-dimensional domain is such, so non-noetherian almost perfect domains form a class of nice … Read more

## On GCD and LCM of elements in integral domain with Krull-dimension 1

Let R be an integral domain with Krull dimension 1. If 0≠a∈R is such that for every b∈R , the ideal Ra∩Rb is principal , then is it true that for every b∈R, the ideal Ra+Rb is also principal ? It is clear that if 0≠a,b and Ra+Rb=Rd , then Ra∩Rb=R(ab/d). So if we know … Read more

## Is every universally catenary ring a going-between ring?

This question asks for the necessity of a noetherian hypothesis in a certain relation between properties of rings concerning chains of prime ideals. We use the following definitions. A ring $R$ is called universally catenary if every $R$-algebra of finite type is catenary. (Note that $R$ need not be noetherian.) A ring $R$ is called … Read more

## Algebraic characterization of commutative rings of Krull dimension 1,2, or 3

A commutative ring R (with 1) is 0-dimensional if and only if R/√0 is von Neumann regular. Besides this result, there is a wealth of information about the algebraic structure of zero-dimensional rings. I could not find any information about the algebraic structure of rings of higher Krull dimension. If the problem for general commutative … Read more

## Catenarity and epimorphisms of rings

Let R be a commutative ring. The following are well-known: If R is catenary and a⊆R is an ideal, then R/a is catenary. If R is catenary and S⊆R is a subset, then S−1R is catenary. This means that catenarity is preserved along two special kinds of epimorphisms of commutative rings. In view of this … Read more

## Generalization of Krull dimension for commutative rings

In the paper How to construct huge chains of prime ideals in power series rings by B. Kang and P. Toan the Krull dimension of a commutative ring with 1 is defined as follows: Let R be a commutative ring and C be an arbitrary chain of prime ideals in R. Then length of C … Read more

## How bad does a ring have to be for a failure of “going-in-between”?

Let A⊂B be an integral extension of commutative unital rings. Let p0⊂p1⊂p2 be a saturated chain of primes in A of length 2. Suppose q0,q2 lie over p0,p2, and q0⊂q2. Is there necessarily a q1 satisfying q0⊂q1⊂q2 and lying over p1? It seems to me the answer is clearly yes if the rings A,B are … Read more

## Should Krull dimension be a cardinal?

A totally ordered finite set P0⊊P1⊊⋯⊊Pn of prime ideals of a ring A is said to be a chain of length n. As is well known, the supremum of the lengths of such chains is called the Krull dimension dim(A) of the ring A. If the lengths of these chains are not bounded, the ring … Read more