<i>QF</i>-3 rings with zero singular ideal

Colby, Robert R.; Rutter, Edgar

doi:10.2140/pjm.1969.28.303

Cited by 23 publications

(14 citation statements)

References 8 publications

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“…(3) implies (6). It follows from [13, Proposition 2.4] that since E(RR) is projective, it is finitely generated.…”

Section: R R Colby and E A Rutter Jrmentioning

confidence: 95%

“…It is clear that (1) and (2) each imply (3). We shall first prove that (3), (4), and (5) each imply (6) and then show that (7) follows from (6). It is known that (7) implies all the others (see [25] or [18]).…”

Section: R R Colby and E A Rutter Jrmentioning

confidence: 99%

“…Thus ß contains no infinite set of orthogonal idempotents. Furthermore, we may assume that these ideals are indexed in such a way that each left ideal which appears in the above decomposition is isomorphic to exactly one of (4) implies (6). Let Re with e2 = e be the faithful injective left ideal of ß.…”

Section: R R Colby and E A Rutter Jrmentioning

confidence: 99%

“…Since Re is faithful we may identify R with a subring of R in the usual manner and this implies that R contains no infinite set of orthogonal idempotents since Ê does not. Thus R satisfies condition (6). (5) implies (6).…”

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confidence: 97%

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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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“…(3) implies (6). It follows from [13, Proposition 2.4] that since E(RR) is projective, it is finitely generated.…”

Section: R R Colby and E A Rutter Jrmentioning

confidence: 95%

Section: R R Colby and E A Rutter Jrmentioning

confidence: 99%

Section: R R Colby and E A Rutter Jrmentioning

confidence: 99%

mentioning

confidence: 97%

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Generalizations of 𝑄𝐹-3 algebras

Colby¹,

Rutter²

1971

Trans. Amer. Math. Soc.

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“…If a module M is f.g. projective, co-f.g. injective and faithful, then M will be called a '-module for convenience. If a ring A is left ßF-3, then a minimal faithful left A -module is a *-module, and conversely if A has a left *-module, then A is left ßF-3 (see [2,Theorem I]). Let A D B 3 lA be a ring extension.…”

mentioning

confidence: 99%

Frobenius extensions of QF-3 rings

Kitamura¹

1980

Proc. Amer. Math. Soc.

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Abstract. We investigate the inheritance of QF-3 property for ring extensions, mainly, for Frobenius extensions. Let A be a ring with identity. It is proved that a group ring A[G] of A with a finite group G is left ßF-3 iff A is left QF-3 and that in case A is a. G-Galois extension of the fixed subring A ° relative to a finite group G of ring automorphism of A, A is left QF-3 iff A G is left QF-3.

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Rings of quotients of perfect rings

Storrer

1971

Math Z

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QF-3 rings with zero singular ideal

Cited by 23 publications

References 8 publications

Generalizations of 𝑄𝐹-3 algebras

Generalizations of 𝑄𝐹-3 algebras

Frobenius extensions of QF-3 rings

Rings of quotients of perfect rings

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