The European Solar Telescope (EST) is a project aimed at studying the magnetic connectivity of the solar atmosphere, from the deep photosphere to the upper chromosphere. Its design combines the knowledge and expertise gathered by the European solar physics community during the construction and operation of state-of-the-art solar telescopes operating in visible and near-infrared wavelengths: the Swedish 1m Solar Telescope, the German Vacuum Tower Telescope and GREGOR, the French Télescope Héliographique pour l’Étude du Magnétisme et des Instabilités Solaires, and the Dutch Open Telescope. With its 4.2 m primary mirror and an open configuration, EST will become the most powerful European ground-based facility to study the Sun in the coming decades in the visible and near-infrared bands. EST uses the most innovative technological advances: the first adaptive secondary mirror ever used in a solar telescope, a complex multi-conjugate adaptive optics with deformable mirrors that form part of the optical design in a natural way, a polarimetrically compensated telescope design that eliminates the complex temporal variation and wavelength dependence of the telescope Mueller matrix, and an instrument suite containing several (etalon-based) tunable imaging spectropolarimeters and several integral field unit spectropolarimeters. This publication summarises some fundamental science questions that can be addressed with the telescope, together with a complete description of its major subsystems.

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In this paper, the rings which have a torsion theory T with associated torsion radical t such that R/t(R) has a minimal r-torsionfree cogenerator are studied. When T is the trivial torsion theory these are precisely the left QF-3 rings. For T = TL, the Lambek torsion theory, this class of rings is wider but, with an additional hypothesis on TL it is shown that if R has this property with respect to the Lambek torsion theory on both sides, then R is a (left and right) QF-3 ring. The results obtained are applied to get new characterizations of QF-3 rings with the ascending chain condition on left annihilators. A ring R is called left QF-3 if it has a minimal faithful left i?-module and left QF-3' if the injective envelope E(RR) is a torsionless module. These rings have been the object of extensive study (for example in [3, 9, 11, 12, 13, 16]) and, recently, Baccella has obtained in [2] structure results for the class of nonsingular, finite-dimensional QF-3 rings. In the present paper we consider a (hereditary) torsion theory r in i?-mod with associated torsion radical t and the property that R/t{R) has a minimal r-torsionfree cogenerator X, in the sense that X is a r-torsionfree module which cogenerates R/t(R) and is a direct summand of every r-torsionfree cogenerator of R/t(R). When r is the trivial torsion theory in which all i2-modules are r-torsionfree, this property defines left QF-3 rings. The class of rings which have this property with respect to the Lambek torsion theory TL is wider than the class of left QF-3 rings but when TL is strongly semiprime and R has this property for the Lambek torsion theory on both sides, This work was partially supported by the CAICYT (0784-84).

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A ring R is called left QF-3 if it has a minimal faithful left R-module. The endomorphism ring of such a module has been recently studied in [7], where conditions are given for it to be a left PF ring or a QF ring. The object of the present paper is to study, more generally, when the endomorphism ring of a 2-quasi-projective module over any ring R is left QF-3. Necessary and sufficient conditions for this to happen are given in Theorem 2. An useful concept in this investigation is that of a QF-3 module which has been introduced in [11]. If M is a finitely generated quasi-projective module and o [M] denotes the category of all modules isomorphic to submodules of modules generated by M, then we show that End( R M) is a left QF-3 ring if and only if the quotient module of M modulo its torsion submodule (in the torsion theory of o [M] canonically defined by M) is a QF-3 module (Corollary 4). Finally, we apply these results to the study of the endomorphism ring of a minimal faithful R-module over a left QF-3 ring, extending some of the results of [7].Throughout this paper R denotes an associative ring with identity, and /?-mod denotes the category of left R-modules. If M is a module, then we will say that a module N is M-generated (M-cogenerated) if it is a quotient (resp. a submodule) of a direct sum Af (/) (resp. direct product M 1 ) of copies of M. If N is M-cogenerated, then we will also say that N is M-torsionless and that M is a N-cogenerator. The full subcategory of /?-mod consisting of the submodules of M-generated modules will be denoted by o [M]; it is a locally finitely generated Grothendieck category [11]. We recall that a module N is M-projective (M-injective) if, for every quotient module (resp. submodule) X of M, the homomorphism Hom R (N, M)->Hom R (N, X) (resp. Hom R (M, N)-*Wom R {X, N)) is an epimorphism and, in particular, M is quasi-projective when it is M-projective. M is a projective object of a [M] precisely when it is Z-quasi-projective, that is, M (/) is quasi-projective for each set /. The largest M-generated submodule of a module X will be denoted by X M . E(N) will stand for an injective envelope of N in R-mod; if N belongs to o [M], then its injective envelope in this category is precisely E(N) M . A module is called finitely cogenerated (FC for short) if it has a finitely generated essential socle. When R R is injective and finitely cogenerated, R is said to be a left PF ring. The endomorphism ring of a module M will be denoted by S = End( R M) and we will use the convention of writing endomorphisms opposite scalars. We refer the reader to [2] and [6] for all the ring-theoretic notions used in the text.In [11], a module M is called a QF-3 module if there exists a minimal M-cogenerator t Work partially supported by the CAICYT (0784-84).Glasgow Math. J. 30 (1988) 215-220.https://www.cambridge.org/core/terms. https://doi