Abstract
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〈 X 1,…, X n 〉 residually Jaffard (resp., totally Jaffard) when R( X 1,…, X n ) is ? We construct a three-dimensional local ring R such that R( X 1,…, X n ) is totally Jaffard (and hence, residually Jaffard) whereas R〈 X 1,…, X n 〉 is not residually Jaffard (and hence, not totally Jaffard).
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