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Modules semicocritical with respect to a torsion theory and their applications. (English) Zbl 0603.16022
The paper covers four main topics: $$\tau$$-semicocritical modules, $$\tau$$- primitive ideals, application to $$\tau$$-composition series, and applications to endomorphism rings. The author indicates that the first two sections of the paper are devoted to generalizing the basic theory of A. Boyle and E. Feller [Commun. Algebra 11, 1643-1674 (1983; Zbl 0529.16015)] from modules over rings with Krull dimension to modules that are torsionfree with respect to a hereditary torsion theory $$\tau$$. In particular, $$\tau$$-semicocritical and $$\tau$$-primitive ideals are defined based on the corresponding Krull dimension concepts. ($$\tau$$- primitive ideals were first defined and considered by S. Riley and this reviewer [in Proc. Am. Math. Soc. 82, 527-532 (1981; Zbl 0474.16018)].) The author uses the concept of a $$\tau$$-semicocritical module to obtain an ascending chain of submodules of a torsionfree module which he calls the $$\tau$$-semicocritical socle series. He then obtains results by comparing this series to the standard $$\tau$$-cocritical socle series of a torsionfree module. In particular, when R has DCC on $$\tau$$- closed left ideals, he shows that the $$\tau$$-semicocritical socle series coincides with the $$\tau$$-cocritical socle series for any torsionfree module. Next, the author shows that for a ring with DCC on $$\tau$$-closed left ideals, each indecomposable torsionfree injective module is associated with a minimal $$\tau$$-primitive ideal of the ring. This shows that the minimal $$\tau$$-primitive ideals play an important role in the study of torsionfree injective modules and their rings of endomorphisms.
In the last two sections of the paper, the author applies these results to obtain results on the $$\tau$$-composition series of a torsionfree module and results on the endomorphism ring of a torsionfree module over a ring with DCC on $$\tau$$-closed left ideals. If R has DCC on $$\tau$$- closed left ideals, the author shows that a nonzero torsionfree module has a finite $$\tau$$-semicocritical socle series. Under these conditions, an important idea introduced in the paper is that of linkage of a $$\tau$$- primitive ideal of the ring to a nonzero torsionfree module M. If D is a $$\tau$$-primitive ideal, then D is said to be linked to M at the i-th layer if D annihilates some nonzero submodule of the i-th factor of the $$\tau$$-semicocritical socle series for M. This concept of linkage is then used to study the endomorphism ring of a torsionfree module over a ring with DCC on $$\tau$$-closed left ideals. The main result gives a linkage condition for the endomorphism ring of an indecomposable torsionfree injective module to be a division ring.
Theorem. Let R have DCC on $$\tau$$-closed left ideals, let I be an indecomposable torsionfree injective module, and let D be the minimal $$\tau$$-primitive ideal associated with I. If $$0\subset Sc(I)\subset...Sc^{n-1}(I)\subset Sc^ n(I)=I$$ is the $$\tau$$- semicocritical socle series of I, then the following are equivalent: (1) $$End_ R(I)$$ is a division ring. (2) $$Hom_ R(I/K,I)=0$$ for each nonzero $$K\subseteq I$$. (3) $$Hom_ R(I/Sc^ i(I),I)=0$$ for each $$i\geq 1$$. (4) D is linked to I only at the first layer.
Reviewer: P.E.Bland

##### MSC:
 16S90 Torsion theories; radicals on module categories (associative algebraic aspects) 16D50 Injective modules, self-injective associative rings 16S50 Endomorphism rings; matrix rings 16P60 Chain conditions on annihilators and summands: Goldie-type conditions 16Dxx Modules, bimodules and ideals in associative algebras
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