Abstract
Let RT be an extension of integral domains, X be an indeterminate over T, and R(X) and T(X) be polynomial rings. Then R � T is said to be LCM-stable if (aR bR)T = bT for all 0 6 a,b 2 R. Let wA be the so-called w-operation on an integral domain A. In this paper, we introduce the notions of w(e)- and w-LCM stable extensions: (i) RT is w(e)-LCM-stable if bR)T)wT = aT bT for all 0 6 a,b 2 R and (ii) RT is w-LCM-stable if ((aR bR)T)wR = (aT bT)wR for all 0 6 a,b 2 R. We prove that LCM-stable extensions are both w(e)-LCM- stable and w-LCM-stable. We also generalize some results on LCM-stable extensions. Among other things, we show that if R is a Krull domain (resp., PvMD), then RT is w(e)-LCM-stable (resp., w-LCM-stable) if and only if R(X) � T(X) is w(e)-LCM-stable (resp., w-LCM-stable).
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