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On almost excellent extensions. (English) Zbl 0851.16024

Ring extensions play an important role in the study of rings and modules. The paper under review concerns a recent generalization, introduced by the author, of an interesting ring extension – “excellent extension”, which is a restricted finite normalizing extension.
A unitary ring extension \(S \supset R\) is called a finite normalizing extension if there is a finite subset \(\{s_1,s_2,\dots,s_n\}\subset S\) such that \(S=\sum^n_{i=1} s_iR\) and \(s_iR=Rs_i\) for all \(i=1,2,\dots,n\). Finite normalizing extension has been used to study the structure of group rings and group algebras by several authors. Later, D. S. Passman [The Algebraic Structure of Group Rings, Wiley-Interscience, New York (1977; Zbl 0368.16003)] and L. Bonami [On the Structure of Skew Group Rings, Algebra Berichte 48, Verlag R. Fischer, München (1984; Zbl 0537.16005)] introduced a special kind of finite normalizing extension, i.e. excellent extension. A ring \(S\) is called an excellent extension of ring \(R\) if (1) \(S\) is right \(R\)-projective, that is, \(N_R\) is a direct summand of \(M_R\) always implies that \(N_S\) is a direct summand of \(M_S\) for any \(S\)-module \(M\) and \(N\); (2) \(S\) is a free normalizing extension of \(R\), that is, \(S=\sum^n_{i=1} s_iR\) is a finite normal extension and \(S\) is free with basis \(\{s_1=1,s_2,\dots,s_n\}\) both as a right \(R\)-module and a left \(R\)-module. A finite normalizing extension \(S\supset R\) is called an almost excellent extension if (1) \(S\) is right \(R\)-projective; (2) \(S_R\) is projective and \(_RS\) is flat. Clearly, this is a generalization of excellent extension.
Let \(pd(-)\), \(id(-)\), \(fd(-)\) denote the projective, injective and flat dimensions of a module, and \(rD(-)\) and \(wD(-)\) denote the right global and weak global dimensions of a ring. It is proved, in this paper, that the following theorems hold for an almost excellent extension \(S \supset R\): (i) if \(rD(R)<\infty\) then \(rD(R)=rD(S)\); (ii) if \(wD(R) < \infty\) then \(wD(R)=wD(S)\); (iii) If one of \(S\) and \(R\) is right FS, right hereditary, right semihereditary, right SI, right GV, right \(S^3 I\) or right almost Artinian (Noetherian), then so is the other. Finally, some results on dual Goldie dimension over a finite normalizing extension are also obtained.
Reviewer: H.-P.Yu (Emory)

MSC:

16S20 Centralizing and normalizing extensions
16E10 Homological dimension in associative algebras
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