Torsion theories and semihereditary rings

Turnidge, Darrell R.

doi:10.1090/s0002-9939-1970-0255601-1

Cited by 9 publications

(9 citation statements)

References 12 publications

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“…The equivalence of (b) and (c) is clear from Remark (1) following Lemma 3. The equivalence of (a) and (b) will follow immediately from Theorem 1 if we show that the ring hypothesis implies every nonsingular i?-module is R-Ά&t. But this follows from [11,Cor. 2.5] and [11,Theorem 2.1].…”

Section: V-soc (V) But For Any Nonsingular Simple R-module S Hom β (Smentioning

confidence: 95%

“…Then by (i), K= f/φ V, where U is finitely generated and As a corollary, we have the following result for left hereditary rings: COROLLARY 1. Let R be a left hereditary ring whose maximal quotient ring R Q (see [3], [11]) is R-flat. Then the following statements are equivalent for any finitely generated nonsingular Rmodule N:…”

Section: But Then F/h Does Not Have a Finitely Generated Singular Submentioning

confidence: 99%

“…If R is a right semi-hereditary ring having a maximal left quotient ring Q (see (3], [11]), which is a two-sided quotient ring of R, then the following statements are equivalent:…”

Section: Corollary 3 If Z(r) -0 and If Every Closed Submodule Of A Fmentioning

confidence: 99%

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The singular submodule of a finitely generated module splits off

Fuelberth¹,

Teply²

1972

Pacific J. Math.

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Section: V-soc (V) But For Any Nonsingular Simple R-module S Hom β (Smentioning

confidence: 95%

Section: But Then F/h Does Not Have a Finitely Generated Singular Submentioning

confidence: 99%

See 1 more Smart Citation

The singular submodule of a finitely generated module splits off

Fuelberth¹,

Teply²

1972

Pacific J. Math.

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“…properties of F(A) and those of A where (F, λ) is an /-functor on Mod-A, much in the same manner as Turnidge [7]. LEMMA …”

Section: F(m) ~ ^U F(n)mentioning

confidence: 86%

Cotorsion theories

Bronn¹

1973

Pacific J. Math.

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“…The classical ring of quotients Q(R) with respect to all non zero elements of R is the maximal ring of quotients in the sense of Lambek-Utumi and it is flat as a right and left R module. R is therefore right and left semihereditary (Turnidge (1970)). It follows further that every ring S between R and Q(R) is flat as a right and left R module and the injection R into S is an epimorphism in the category of rings.…”

Section: H H Brungs (Received 27 November 1972)mentioning

confidence: 96%

Filters and overrings

Brungs

1975

J. Aust. Math. Soc.

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Communicated by G. E. WallLet R be an integral domain. It is well known (see Lambek (1971), Stenstrom (1971)), that idempotent filters of right ideals, torsion radicals and torsion theories are in one-to-one correspondence, but that different idempotent filters !F of right ideals may lead to the same rings of quotiens R?. We have always R cz R? c Qmax(R)-Given this situation one can ask a number of questions. For example: Describe all different idempotent filters for a given ring. Determine all different rings of quotients. When do different filters lead to the same ring of quotients? When are all rings between R and Q maji (R) of the form RgP. When is every R? of the form RS~l, where S is an Ore system? Some of the problems mentioned above are easier to handle if it is possible to use a localizing procedure. This is described in section 1, and one can apply it for example to all noetherian commutative domains (where one knows all different filters). We then consider noncommutative Krull domains. These rings are a generalization of Krull domains, and it is possible to determine all their rings of quotients.It is known (see for example Gilmer and Ohm (1964)) that every ring between a commutative noetherian integral domain R and its field of quotients is always a ring of quotients with respect to some multiplicative system if and only if R is a Dedekind domain with torsion class group. We will give a similar condition for the semigroup of divisorial ideals of a non commutative Krull domain to insure that every ring of quotients with respect to some torsion theory is a ring of quotients with respect to some Ore system.We add some related results about principal ideal domains and Bezont domains in the final section.We will use the word filter instead of idempotent filter and we will use the definition in Stenstrom (1971)

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Torsion theories and semihereditary rings

Cited by 9 publications

References 12 publications

The singular submodule of a finitely generated module splits off

The singular submodule of a finitely generated module splits off

Cotorsion theories

Filters and overrings

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