krull-dimension

Catenarity of monoid algebras

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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

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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

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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

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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?

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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

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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

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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

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

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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