Details:
Title  The Gr\"obner ring conjecture in the lexicographic order case.  Author(s)  Ihsen Yengui  Type  Article in Journal  Abstract  We prove that a valuation domain V has Krull dimension ≤1 if and only if, for any n, fixing the lexicographic order as monomial order in V[X1,…,Xn], for every finitely generated ideal I of V[X1,…,Xn], the ideal generated by the leading terms of the elements of I is also finitely generated. This proves the Gröbner ring conjecture in the lexicographic order case. The proof we give is both simple and constructive. The same result is valid for Prüfer domains. As a “scoop”, contrary to the common idea that Gröbner bases can be computed exclusively on Noetherian ground, we prove that computing Gröbner bases over R[X1,…,Xn], where R is a Prüfer domain, has nothing to do with Noetherianity, it is only related to the fact that the Krull dimension of R is ≤1.  Keywords  Bezout domain, Valuation domain, Semihereditary ring, Gröbner ring conjecture, Constructive mathematics  ISSN  00255874; 14321823/e 
URL 
http://link.springer.com/article/10.1007%2Fs002090131197y 
Language  English  Journal  Math. Z.  Volume  276  Number  12  Pages  261265  Publisher  Springer, Berlin/Heidelberg  Year  2014  Edition  0  Translation 
No  Refereed 
No 
