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**Galois connections between lattices of preradicals induced by ring epimorphisms.**
*(English)*
Zbl 1453.16032

The authors introduced the concept of a Galois connection between the lattices of predradicals \(R\)-pr and \(S\)-pr of two rings \(R\) and \(S\) induced by any adjoint pair of functors between the categories in an earlier paper [the authors, Appl. Categ. Struct. 24, No. 3, 241–268 (2016; Zbl 1345.16030)]). The paper under review continues this study. For the Galois connection \(\langle \phi , \psi \rangle\) induced by the adjoint pair \(\langle F , G \rangle\) of functors, several results concerning preradicals are found. Let \(R\) and \(S\) be associative rings with identity and let \(\tau\) be any preradical over \(R\) and \(\sigma\) any preradical over \(S\). Then the pretorsion-free class associated with \(\phi (\tau)\) and the pretorsion class associated with \(\psi (\sigma )\) are described explicitly. It is also shown that \(\phi\) preserves idempotency and \(\psi\) preserves radicals.

If \(f:A \rightarrow B\) is a ring homomorphism, then two pairs of adjoint functors \(\langle F,G \rangle\) and \(\langle G,H \rangle\) exist with induced Galois connections \(\langle \phi , \psi \rangle \) and \(\langle \zeta , \xi \rangle\) respectively. If \(\tau\) is a preradical on \(A\) and \(N\) is a \(B\)-submodule, then \(\psi(\tau)(N)\) is shown to be the least \(B\)-submodule containing \(\tau(_{A}N)\) and \(\xi(\tau)(N)\) is shown to be the greatest \(B\)-submodule of \(N\) that is contained in \(\tau(_{A}N)\). This leads to a description of the pretorsion and pretorsion-free classes that correspond to the induced Galois connections.

Whereas the earlier paper of the authors [loc. cit.] focused on the case where \(S\) is a quotient ring, a slightly more general setting is considered here, viz. the case where \(f:A \rightarrow B\) is a ring epimorphism. In this case, it is shown that \(\psi (\sigma)\) is the greatest and \(\zeta(\sigma)\) is the least extension of any preradical \(\sigma\) on \(B\) and it follows that \(\phi\) and \(\xi\) are surjective and \(\psi\) and \(\zeta\) are injective. The class of all extensions of any preradical \(\sigma\) on \(A\) is characterized. If \(\tau\) is an idempotent preradical (resp. radical) on \(B\), then it is proved that the interval \([\zeta(\tau), \psi(\tau)]\) is closed under products (resp. coproducts). Characterizations are also found for the equalities \(\phi = \xi\) and \(\psi = \zeta\) to be true.

If \(f:A \rightarrow B\) is a ring homomorphism, then two pairs of adjoint functors \(\langle F,G \rangle\) and \(\langle G,H \rangle\) exist with induced Galois connections \(\langle \phi , \psi \rangle \) and \(\langle \zeta , \xi \rangle\) respectively. If \(\tau\) is a preradical on \(A\) and \(N\) is a \(B\)-submodule, then \(\psi(\tau)(N)\) is shown to be the least \(B\)-submodule containing \(\tau(_{A}N)\) and \(\xi(\tau)(N)\) is shown to be the greatest \(B\)-submodule of \(N\) that is contained in \(\tau(_{A}N)\). This leads to a description of the pretorsion and pretorsion-free classes that correspond to the induced Galois connections.

Whereas the earlier paper of the authors [loc. cit.] focused on the case where \(S\) is a quotient ring, a slightly more general setting is considered here, viz. the case where \(f:A \rightarrow B\) is a ring epimorphism. In this case, it is shown that \(\psi (\sigma)\) is the greatest and \(\zeta(\sigma)\) is the least extension of any preradical \(\sigma\) on \(B\) and it follows that \(\phi\) and \(\xi\) are surjective and \(\psi\) and \(\zeta\) are injective. The class of all extensions of any preradical \(\sigma\) on \(A\) is characterized. If \(\tau\) is an idempotent preradical (resp. radical) on \(B\), then it is proved that the interval \([\zeta(\tau), \psi(\tau)]\) is closed under products (resp. coproducts). Characterizations are also found for the equalities \(\phi = \xi\) and \(\psi = \zeta\) to be true.

Reviewer: Frieda Theron (Potchefstroom)

### MSC:

16S90 | Torsion theories; radicals on module categories (associative algebraic aspects) |

06A15 | Galois correspondences, closure operators (in relation to ordered sets) |

18A40 | Adjoint functors (universal constructions, reflective subcategories, Kan extensions, etc.) |

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

### Citations:

Zbl 1345.16030
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\textit{R. Fernández-Alonso} and \textit{J. Magaña}, J. Algebra Appl. 19, No. 3, Article ID 2050045, 17 p. (2020; Zbl 1453.16032)

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### References:

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[7] | Fernández-Alonso, R., Raggi, F., Ríos, J., Rincón, H. and Signoret, C., The lattice structure of preradicals II. Partitions, J. Algebra Appl.1(2) (2002) 201-214. · Zbl 1034.16035 |

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