Strongly FP-injective and strongly flat functors

2016 ◽  
Vol 135 ◽  
pp. 133-149 ◽  
Author(s):  
Lixin Mao
Keyword(s):  
Strongly Flat

Strongly Flat Covers

2002 ◽  
Vol 66 (2) ◽  
pp. 276-294 ◽  
Author(s):  
S. Bazzoni ◽  
L. Salce
Keyword(s):  
Strongly Flat

scholarly journals FLAT RING EPIMORPHISMS OF COUNTABLE TYPE

2019 ◽  
Vol 62 (2) ◽  
pp. 383-439 ◽  
Author(s):  
LEONID POSITSELSKI
Keyword(s):  
Associative Ring ◽  
Abelian Category ◽  
Projective Dimension ◽  
Countable Base ◽  
Linear Topology ◽  
Countable Type ◽  
Flat Ring ◽  
Abelian Categories ◽  
Induced Topology ◽  
Strongly Flat

AbstractLet R→U be an associative ring epimorphism such that U is a flat left R-module. Assume that the related Gabriel topology $\mathbb{G}$ of right ideals in R has a countable base. Then we show that the left R-module U has projective dimension at most 1. Furthermore, the abelian category of left contramodules over the completion of R at $\mathbb{G}$ fully faithfully embeds into the Geigle–Lenzing right perpendicular subcategory to U in the category of left R-modules, and every object of the latter abelian category is an extension of two objects of the former one. We discuss conditions under which the two abelian categories are equivalent. Given a right linear topology on an associative ring R, we consider the induced topology on every left R-module and, for a perfect Gabriel topology $\mathbb{G}$, compare the completion of a module with an appropriate Ext module. Finally, we characterize the U-strongly flat left R-modules by the two conditions of left positive-degree Ext-orthogonality to all left U-modules and all $\mathbb{G}$-separated $\mathbb{G}$-complete left R-modules.

Strongly Flat and Condition (P) Covers of Acts Over Monoids

2010 ◽  
Vol 38 (12) ◽  
pp. 4520-4530 ◽  
Author(s):  
R. Khosravi ◽  
M. Ershad ◽  
M. Sedaghatjoo
Keyword(s):  
Strongly Flat

On Monoids for Which all Condition (P) Acts are Strongly Flat

2009 ◽  
Vol 37 (1) ◽  
pp. 234-241 ◽  
Author(s):  
Husheng Qiao ◽  
Fang Li
Keyword(s):  
Strongly Flat

Pullback-Flat Acts are Strongly Flat

1991 ◽  
Vol 34 (4) ◽  
pp. 456-461 ◽  
Author(s):  
Sydney Bulman-Fleming
Keyword(s):  
Interpolation Condition ◽  
Strong Flatness ◽  
Strongly Flat

AbstractLet 5 be a monoid. A right S-system A is called strongly flat if the functor A ⊗ — (from the category of left S-systems into the category of sets) preserves pullbacksand equalizers. (This concept arises in B. Stenström, Math. Nachr. 48(1971), 315-334 under the name weak flatness). The main result of the present paper is a proof that for A to be strongly flat it is in fact sufficient that A ⊗ — preserve only pullbacks. The approach taken is to develop an "interpolation" condition for pullback-preservation, and then to show its equivalence to Stenström's conditions for strong flatness.

scholarly journals COVERS FOR S-ACTS AND CONDITION (A) FOR A MONOID S

2014 ◽  
Vol 57 (2) ◽  
pp. 323-341
Author(s):  
ALEX BAILEY ◽  
VICTORIA GOULD ◽  
MIKLÓS HARTMANN ◽  
JAMES RENSHAW ◽  
LUBNA SHAHEEN
Keyword(s):  
Chain Condition ◽  
Projective Cover ◽  
Descending Chain Condition ◽  
Strongly Flat ◽  
Ascending Chain Condition

AbstractA monoid S satisfies Condition (A) if every locally cyclic left S-act is cyclic. This condition first arose in Isbell's work on left perfect monoids, that is, monoids such that every left S-act has a projective cover. Isbell showed that S is left perfect if and only if every cyclic left S-act has a projective cover and Condition (A) holds. Fountain built on Isbell's work to show that S is left perfect if and only if it satisfies Condition (A) together with the descending chain condition on principal right ideals, MR. We note that a ring is left perfect (with an analogous definition) if and only if it satisfies MR. The appearance of Condition (A) in this context is, therefore, monoid specific. Condition (A) has a number of alternative characterisations, in particular, it is equivalent to the ascending chain condition on cyclic subacts of any left S-act. In spite of this, it remains somewhat esoteric. The first aim of this paper is to investigate the preservation of Condition (A) under basic semigroup-theoretic constructions. Recently, Khosravi, Ershad and Sedaghatjoo have shown that every left S-act has a strongly flat or Condition (P) cover if and only if every cyclic left S-act has such a cover and Condition (A) holds. Here we find a range of classes of S-acts $\mathcal{C}$ such that every left S-act has a cover from $\mathcal{C}$ if and only if every cyclic left S-act does and Condition (A) holds. In doing so we find a further characterisation of Condition (A) purely in terms of the existence of covers of a certain kind. Finally, we make some observations concerning left perfect monoids and investigate a class of monoids close to being left perfect, which we name left$\mathcal{IP}$a-perfect.

Some Conditions on Monoids for Which Condition (P) Acts Are Strongly Flat

2004 ◽  
Vol 32 (12) ◽  
pp. 4795-4807 ◽  
Author(s):  
Qiao HuSheng ◽  
Li Lian
Keyword(s):  
Strongly Flat

Flat and strongly flat s-systems

1992 ◽  
Vol 20 (9) ◽  
pp. 2553-2567 ◽  
Author(s):  
Sydney Bulman Fleming
Keyword(s):  
Strongly Flat
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