On projective modules over semi-hereditary rings

Albrecht, Felix

doi:10.1090/s0002-9939-1961-0126470-x

Cited by 25 publications

(27 citation statements)

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“…Similar techniques are applied to show that every projective left module over a right semihereditary ring is a direct sum of finitely generated modules. The corresponding result for left modules over left semihereditary rings was proved by Albrecht [1] and our result, like his, generalizes Kaplansky's treatment of the commutative case and depends heavily on Kaplansky's reduction of the problem [23].…”

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confidence: 70%

“…Kaplansky used these results to deduce, among other things, that if R is a commutative semihereditary ring then every projective .R-module is a direct sum of finitely generated modules. This result was generalized by Albrecht [ 1 ] who used Kaplansky's reduction to give a very easy proof that every projective right Ä-module over a right semihereditary ring is the direct sum of finitely generated projective modules. We close this section with a different generalization of Kaplansky's results.…”

Section: R R Colby and E A Rutter Jrmentioning

confidence: 99%

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Generalizations of 𝑄𝐹-3 algebras

Colby¹,

Rutter²

1971

Trans. Amer. Math. Soc.

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Abstract. This paper consists of three parts. The first is devoted to investigating the equivalence and left-right symmetry of several conditions known to characterize finite dimensional algebras which have a unique minimal faithful representation-QF-3 algebras-in the class of left perfect rings. It is shown that the following conditions are equivalent and imply their right-hand analog: R contains a faithful S-injective left ideal, R contains a faithful LT-projective injective left ideal; the injective hulls of projective left Ä-modules are projective, and the projective covers of injective left .R-modules are injective. Moreover, these rings are shown to be semiprimary and to include all left perfect rings with faithful injective left and right ideals.The second section is concerned with the endomorphism ring of a projective module over a hereditary or semihereditary ring. More specifically we consider the question of when such an endomorphism ring is hereditary or semihereditary.In the third section we establish the equivalence of a number of conditions similar to those considered in the first section for the class of hereditary rings and obtain a structure theorem for this class of hereditary rings. The rings considered are shown to be isomorphic to finite direct sums of complete blocked triangular matrix rings each over a division ring.Thrall [36] called an algebra A of finite rank (left) QF-3 if it has a unique minimal faithful left module, that is, a unique (up to isomorphism) module M with the property that no proper direct summand is faithful. It is not difficult to verify the equivalence of the following statements.(1) A is (left) QF-3. Moreover, the right-hand analogue of each of these conditions is easily established by forming duals with respect to the field so that for algebras being QF-3 is a two-sided property and distinctions between left and right are unnecessary.It is apparent from the above list that there are several possible ring theoretic generalizations of QF-3 algebras and this paper is primarily devoted to resolving certain questions which arise naturally in connection with such extensions. Key words and phrases. QF-3 ring, left perfect ring, unique minimal faithful module, faithful S-injective left ideal, faithful LT-projective injective left ideal, injective hull projective, projective cover injective duality, projective module, endomorphism ring, hereditary ring, semihereditary ring.

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confidence: 70%

Section: R R Colby and E A Rutter Jrmentioning

confidence: 99%

Generalizations of 𝑄𝐹-3 algebras

Colby¹,

Rutter²

1971

Trans. Amer. Math. Soc.

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“…On the other hand, we know by [1] that Ker π is a direct sum of finitely generated projective modules, say Ker π = T P t . We claim that there exists an n ∈ N such that R n contains a local direct summand H X h with |H| |T | and X h = 0 for every h ∈ H .…”

Section: Definitionmentioning

confidence: 99%

“…As R R is hereditary, both P , Q are projective and therefore, they are direct sums of finitely generated projective modules by [1], say,…”

Section: And This Meansmentioning

confidence: 99%

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Hereditary rings with countably generated cotorsion envelope

Asensio

Pusat

2014

Journal of Algebra

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K0 of a semilocal ring

Facchini

Herbera

2000

Journal of Algebra

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Let R be a semilocal ring, that is, R modulo its Jacobson radical J R is artinian. Then K 0 R/J R is a partially ordered abelian group with order-unit, isomorphic to Z n ≤ u , where ≤ denotes the componentwise order on Z n and u is an order-unit in Z n ≤ . Moreover, the canonical projection π: R → R/J R induces an embedding of partially ordered abelian groups with order-unit K 0 π : K 0 R → K 0 R/J R . In this paper we prove that every embedding of partially ordered abelian groups with order-unit G → Z n can be realized as the mapping K 0 π : K 0 R → K 0 R/J R for a suitable hereditary semilocal ring R.

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On projective modules over semi-hereditary rings

Cited by 25 publications

References 1 publication

Generalizations of 𝑄𝐹-3 algebras

Generalizations of 𝑄𝐹-3 algebras

Hereditary rings with countably generated cotorsion envelope

K0 of a semilocal ring

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