The Dual Baer Criterion for non-perfect rings
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Jan Trlifaj
Abstract
Baer’s Criterion for Injectivity is a useful tool of the theory of modules. Its dual version (DBC) is known to hold for all right perfect rings, but its validity for the non-right perfect ones is a complex problem (first formulated by C. Faith [Algebra. II. Ring Theory, Springer, Berlin, 1976]). Recently, it has turned out that there are two classes of non-right perfect rings: (1) those for which DBC fails in ZFC, and (2) those for which DBC is independent of ZFC. First examples of rings in the latter class were constructed in [J. Trlifaj, Faith’s problem on R-projectivity is undecidable, Proc. Amer. Math. Soc. 147 2019, 2, 497–504]; here, we show that this class contains all small semiartinian von Neumann regular rings with primitive factors artinian.
Dedicated to Roger and Sylvia Wiegand, in honor of their 150th birthday
Funding source: Grantová Agentura České Republiky
Award Identifier / Grant number: GACR 17-23112S
Funding statement: Research supported by GAČR 17-23112S.
Acknowledgements
The author thanks the referee for suggesting several simplifications of the presentation. He also thanks Roger Wiegand for discussions related to the commutative noetherian case (cf. Remark 2.5).
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© 2021 Walter de Gruyter GmbH, Berlin/Boston
Artikel in diesem Heft
- Frontmatter
- Approximating pointwise products of quasimodes
- On the group of a rational maximal bifix code
- Hörmander type multiplier theorems on bi-parameter anisotropic Hardy spaces
- Newton’s method for nonlinear stochastic wave equations
- On the classification of Schreier extensions of monoids with non-abelian kernel
- Smoothness filtration of the magnitude complex
- Varieties of nilpotent Lie superalgebras of dimension ≤ 5
- The Dual Baer Criterion for non-perfect rings
- Spectral property of the planar self-affine measures with three-element digit sets
- Spectral properties of certain Moran measures with consecutive and collinear digit sets
- Envelopes of circles and spacelike curves in the Lorentz–Minkowski 3-space
- A polynomial bound for the number of maximal systems of imprimitivity of a finite transitive permutation group
- Well-posedness of backward stochastic partial differential equations with Lyapunov condition
- Free division rings of fractions of crossed products of groups with Conradian left-orders
- Strong uncountable cofinality for unitary groups of von Neumann algebras
- Irreducible holonomy groups and first integrals for holomorphic foliations
- Alexandroff topologies and monoid actions
Artikel in diesem Heft
- Frontmatter
- Approximating pointwise products of quasimodes
- On the group of a rational maximal bifix code
- Hörmander type multiplier theorems on bi-parameter anisotropic Hardy spaces
- Newton’s method for nonlinear stochastic wave equations
- On the classification of Schreier extensions of monoids with non-abelian kernel
- Smoothness filtration of the magnitude complex
- Varieties of nilpotent Lie superalgebras of dimension ≤ 5
- The Dual Baer Criterion for non-perfect rings
- Spectral property of the planar self-affine measures with three-element digit sets
- Spectral properties of certain Moran measures with consecutive and collinear digit sets
- Envelopes of circles and spacelike curves in the Lorentz–Minkowski 3-space
- A polynomial bound for the number of maximal systems of imprimitivity of a finite transitive permutation group
- Well-posedness of backward stochastic partial differential equations with Lyapunov condition
- Free division rings of fractions of crossed products of groups with Conradian left-orders
- Strong uncountable cofinality for unitary groups of von Neumann algebras
- Irreducible holonomy groups and first integrals for holomorphic foliations
- Alexandroff topologies and monoid actions
