Hereditary Noetherian prime rings. III: Infinitely generated projective modules
This is the last paper of a trilogy that brings the theory of projective modules over an HNP ring (= hereditary Noetherian prime ring) \(R\) roughly up to the same level as in the case where \(R\) is a commutative Dedekind domain. It describes the structure of the infinitely generated projective right \(R\)-modules. As in the finitely generated case, which is dealt with in part II [J. Algebra 218, No. 2, 338-372 (1999; Zbl 0946.16017)], the description involves the notion of the genus \(\Psi(P)\) of a projective \(R\)-module \(P\). This is a function defined on \(\{0\}\cup\mathcal W\), where \(\mathcal W\) is a set of representatives of the isomorphism classes of unfaithful simple \(R\)-modules; it is constructed as follows. Let \(Q(R)\) be the simple Artinian classical quotient ring of the HNP ring \(R\). Set \(\Psi(P)_0=Q(R)\)-rank of \(P\otimes_R Q(R)\) = the (finite or infinite) number of simple direct summands in a direct sum decomposition of the semisimple \(Q(R)\)-module \(P\otimes_R Q(R)\). If \(W\in\mathcal W\) with annihilator \(M\), then \(R/M\) is a simple Artinian ring and \(\Psi(P)_W\) is defined to be the \(R/M\)-rank of \(P/PM\). It turns out that if \(P\) and \(P'\) are infinitely generated projective \(R\)-modules, then \(P\cong P'\) if and only if \(\Psi(P)=\Psi(P')\). The main theorem of the paper establishes precisely when a family of cardinal numbers, indexed by \(\{0\}\cup\mathcal W\), is the genus \(\Psi(P)\) of some infinitely generated projective \(R\)-module \(P\); it also provides a construction of \(P\). Very explicit descriptions are obtained in two cases: (1) If \(R\) is any HNP ring and \(P_R\) is uncountably generated projective, then \(P\cong H\oplus\bigoplus_iS^{\alpha_i}_i\), where \(H_R\) is finitely generated projective, the \(\alpha_i\) are infinite cardinal numbers satisfying \(\alpha_1<\alpha_2<\cdots\) with at least one \(\alpha_i\) uncountable, and \(S_1\subset S_2\subset\cdots\) is a finite or countably infinite sequence of rings, each a finitely generated right \(R\)-module with \(R\subseteq S_i\subset Q(R)\). Furthermore, the sequence of pairs \((S_i,\alpha_i)\) is unique. (2) If the HNP ring \(R\) is a classical hereditary order (i.e., finitely generated as a module over a central Dedekind domain), then a countably generated projective right \(R\)-module \(P\) is of the form \(P\cong H\oplus S^{\aleph_0}\), where \(H\) is a finitely generated projective \(R\)-module and \(S\) is a ring with \(R\subseteq S\subset Q(R)\) such that \(S_R\) is finitely generated.
simple Artinian classical quotient rings
scientific article; zbMATH DE number 1423614
scientific article