##
**Reflexive bimodules.**
*(English)*
Zbl 0684.16020

Let R and \(\Delta\) be rings with identity, and let \({}_ RM_{\Delta}\) be a bimodule. The authors define
\[
alglat(_ RM_{\Delta})=\{\alpha \in end(M_{\Delta})| \quad \alpha K\subseteq K\quad for\quad all\quad_ RK\subseteq_ RM\}.
\]
There is a natural mapping from R to \(end(M_{\Delta})\) that sends \(r\in R\) to left multiplication by r; \({}_ RM_{\Delta}\) is called reflexive if this map is a surjection. The definition extends a notion from the theory of operator algebras that was introduced by P. Halmos [J. Lond. Math. Soc., II. Ser. 4, 257- 263 (1971; Zbl 0231.47003)] and has subsequently been studied by many people.

Here, the authors first provide some general lemmata, and then go on to characterize the commutative artinian rings R for which every faithful module is reflexive; they are precisely the QR rings. This leads to a new proof of D. Hadwin and J. Kerr’s characterization of commutative semiprimary rings for which every module is reflexive [Proc. Am. Math. Soc. 103, 1-8 (1988; Zbl 0656.13007)]. The Morita invariance of reflexivity is then established.

For an algebra R over a field K, a module \({}_ RM\) is called reflexive if \({}_ RM_ K\) is reflexive as defined above. It is shown that a finite-dimensional K-algebra is split if and only if each of its simple modules is reflexive, and that if every indecomposable projective R- module is reflexive, then R is split and \(eJ(R)e=0\) for every primitive idempotent e. On the other hand, if R is a split hereditary algebra, then every projective R-module is reflexive. Finally, a characterization of reflexive modules over serial algebras given by J. Habibi and the reviewer [Linear Algebra Appl. 99, 217-223 (1988; Zbl 0638.16013)] is reproven.

Here, the authors first provide some general lemmata, and then go on to characterize the commutative artinian rings R for which every faithful module is reflexive; they are precisely the QR rings. This leads to a new proof of D. Hadwin and J. Kerr’s characterization of commutative semiprimary rings for which every module is reflexive [Proc. Am. Math. Soc. 103, 1-8 (1988; Zbl 0656.13007)]. The Morita invariance of reflexivity is then established.

For an algebra R over a field K, a module \({}_ RM\) is called reflexive if \({}_ RM_ K\) is reflexive as defined above. It is shown that a finite-dimensional K-algebra is split if and only if each of its simple modules is reflexive, and that if every indecomposable projective R- module is reflexive, then R is split and \(eJ(R)e=0\) for every primitive idempotent e. On the other hand, if R is a split hereditary algebra, then every projective R-module is reflexive. Finally, a characterization of reflexive modules over serial algebras given by J. Habibi and the reviewer [Linear Algebra Appl. 99, 217-223 (1988; Zbl 0638.16013)] is reproven.

Reviewer: W.H.Gustafson

### MSC:

16Gxx | Representation theory of associative rings and algebras |

16P10 | Finite rings and finite-dimensional associative algebras |

16W99 | Associative rings and algebras with additional structure |

16L60 | Quasi-Frobenius rings |

16D40 | Free, projective, and flat modules and ideals in associative algebras |

16S50 | Endomorphism rings; matrix rings |

13C05 | Structure, classification theorems for modules and ideals in commutative rings |