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 Rleft angle bracketX1,…,Xnright-pointing angle bracket 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 Rleft angle bracketX1,…,Xnright-pointing angle bracket is not residually Jaffard (and hence, not totally Jaffard).
