Left rigid rings.
Let \(R\) be a ring with \(1\) and \(\mathcal F\) a family of left ideals of \(R\). Then, by definition, \(\text{End}(R^+,\mathcal F)=\{\varphi\in\text{End}(R^+)\mid\forall\,X\in\mathcal F:\varphi(X)\subseteq X\}\). The authors ask when \(\text{End}(R^+,\mathcal F)=R\) and they call \(R\) \(\mathcal F\)-rigid if this is the case. Of particular interest is the case when \(\mathcal F=\mathcal F_\ell\), the family of all left ideals of \(R\). There is no complete solution to this question but there are numerous examples and special cases. Sample results are as follows. Theorem~1. Let \(S\) be a ring with identity, \(\kappa\) a cardinal \(\geq 2\), \(F\) a free \(S\)-module of rank \(\kappa\). Let \(R=\text{End}_S(F)\). Then there exist families \(\mathcal F\), \(\mathcal F_{fin}\) and \(\mathcal F_S\) of left ideals of \(R\) such that (1) \(\text{End}(R^+,\mathcal F)=R\), and if \(\kappa\) is finite, (2) \(\text{End}(R^+,\mathcal F_{fin})=R\), and (3) \(\text{End}_S(R,\mathcal F_S)=R\). Corollary~1. If \(F\) is a field and \(R=\text{End}(F^+)\), then \(\text{End}(R^+,\mathcal F_\ell)=R\). Theorem~4. Let \(R\) be a PID with only finitely many maximal ideals and having a non-zero derivation. Then there is no family \(\mathcal F\) such that \(R=\text{End}(R^+,\mathcal F)\). Corollary~3. Let \(K\) be a field and \(P_1,\dots,P_n\) finitely many maximal ideals of the polynomial ring \(K[x]\). Let \(R\) be the localization of \(K[x]\) at the ideals \(P_i\). Then there is no family \(\mathcal F\) such that \(\text{End}(R^+,\mathcal F)=R\). The authors also deal with polynomial rings in an arbitrary number of indeterminates and with subrings of algebraic number fields. A sample result of the latter is Corollary~5. Let \(S\) be an integrally closed full subring of the finite Galois extension \(F\) of \(\mathbb{Q}\) and \(\mathcal F\) a family of left ideals of \(S\). Then \(E_{\mathcal F}=\text{End}(S,\mathcal F)=S\) or else \(\text{End}(E_{\mathcal F},\widetilde F)=E_{\mathcal F}\) where \(\widetilde F\) is the family of all left ideals of \(E_{\mathcal F}\). On a related theme the authors prove Theorem~11. If \(R=\mathbb{Z}[X]\) is the commutative polynomial ring in a set of indeterminates \(X\) satisfying \(|X|\leq 2^{\aleph_0}\), then there exists an Abelian group \(M\) such that \(R\subseteq M\subseteq\mathbb{Q} R\) and \(\text{End}(M)=R\). Together with known results this establishes that the Theorem~11 holds without restriction on the size of the set of indeterminates.
scientific article; zbMATH DE number 5141714