Theorems in ring theory | Module theory
In abstract algebra, Kaplansky's theorem on projective modules, first proven by Irving Kaplansky, states that a projective module over a local ring is free; where a not-necessary-commutative ring is called local if for each element x, either x or 1 − x is a unit element. The theorem can also be formulated so to characterize a local ring. For a finite projective module over a commutative local ring, the theorem is an easy consequence of Nakayama's lemma. For the general case, the proof (both the original as well as later one) consists of the following two steps: * Observe that a projective module over an arbitrary ring is a direct sum of countably generated projective modules. * Show that a countably generated projective module over a local ring is free (by a "[reminiscence] of the proof of Nakayama's lemma"). The idea of the proof of the theorem was also later used by Hyman Bass to show (under some mild conditions) are free. According to, Kaplansky's theorem "is very likely the inspiration for a major portion of the results" in the theory of semiperfect rings. (Wikipedia).
This lecture is part of an online course on rings and modules. We define projective modules, and give severalexamples of them, including the Moebius band, a non-principal ideal, and the tangent bundle of the sphere. For the other lectures in the course see https://www.youtube.com/playli
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From playlist Global Noncommutative Geometry Seminar (Europe)
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Kaplansky made various related conjectures about group rings, especially for torsion-free groups. For example, the zero divisors conjecture predicts that if K is a field and G is a torsion-free group, then the group ring K[G] has no zero divisors. I will survey what is known about the conj
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This lecture is part of an online course on commutative algebra, following the book "Commutative algebra with a view toward algebraic geometry" by David Eisenbud. We discuss the relation between locally free things (vector bundles) and projective things. In commutative algebra and differe
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From playlist Ring & Module Theory
Giles Gardam: Kaplansky's conjectures
Talk by Giles Gardam in the Global Noncommutative Geometry Seminar (Americas) https://globalncgseminar.org/talks/3580/ on September 17, 2021.
From playlist Global Noncommutative Geometry Seminar (Americas)
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Talk by Pere Are in Global Noncommutative Geometry Seminar (Americas) https://globalncgseminar.org/talks/crossed-products-and-the-atiyah-problem/ on March 19, 2021.
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This lecture is part of an online course on rings and modules. We review basic properties of polynomials over a field, and show that polynomials in any number of variables over a field or the integers have unique factorization. For the other lectures in the course see https://www.youtu
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