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Abstract: We construct an example of a finitely generated ideal of , where is a one-dimensional domain, whose leading terms ideal is not finitely generated. This gives a negative answer to the open question of whether if is a domain with Krull dimension ≤1, then for any finitely generated ideal of , the leading terms ideal of is also finitely generated. Moreover, as a positive part of our answer, we prove that for any one-dimensional domain and any , the ideal of generated by the leading terms of is finitely generated. [Copyright &y& Elsevier]

We propose to give positive answers to the open questions: is R(X,Y) strong S when R(X) is strong S? is R stably strong S (resp., universally catenary) when R[X] is strong S (resp., catenary)? in case R is obtained by a (T,I,D) construction. The importance of these results is due to the fact that this type of ring is the principal source of counterexamples. Moreover, we give an answer to the open questions: is R〈X1,…,Xn〉 residually Jaffard (resp., totally Jaffard) when R(X1,…,Xn) is ? We construct a three-dimensional local ring R such that R(X1,…,Xn) is totally Jaffard (and hence, residually Jaffard) whereas R〈X1,…,Xn〉 is not residually Jaffard (and hence, not totally Jaffard). [Copyright &y& Elsevier]

A multiplicative subset of a commutative ring contains the zero element precisely if the set in question meets every prime ideal. While this form of Krull's Lemma takes recourse to transfinite reasoning, it has recently allowed for a crucial reduction to the integral case in Kemper and the third author's novel characterization of the valuative dimension. We present a dynamical solution by which transfinite reasoning can be avoided, and illustrate this constructive method with concrete examples. We further give a combinatorial explanation by relating the Zariski lattice to a certain inductively generated class of finite binary trees. In particular, we make explicit the computational content of Krull's Lemma. [ABSTRACT FROM AUTHOR]

Abstract: We give a constructive deciphering for a generalization of the Quillen–Suslin theorem due to Maroscia and Brewer & Costa stating that finitely generated projective modules over , where is a Prüfer domain with Krull dimension , are extended from . [Copyright &y& Elsevier]