In abstract algebra, Kaplansky's theorem on projective modules, first proven by Irving Kaplansky, states that a projective module over a local ring is...

Kaplansky's theorem may refer to: Kaplansky's theorem on projective modules Kaplansky's theorem on quadratic forms Kaplansky density theorem This disambiguation...

projective". This fact is easy to prove for finitely generated projective modules. In general, it is due to Kaplansky (1958); see Kaplansky's theorem...

following statement (which is a key step in the proof of Kaplansky's theorem on projective modules): Given an element x ∈ N {\displaystyle x\in N} , there...

over R, or a module of finite type. Related concepts include finitely cogenerated modules, finitely presented modules, finitely related modules and coherent...

1090/s0002-9939-03-07022-9. MR 1963780. Kaplansky's theorem on projective modules Irving Kaplansky at the Mathematics Genealogy Project O'Connor, John J.; Robertson...

flat. Also, Kaplansky's theorem states a projective module over a (possibly non-commutative) local ring is free. Sometimes, whether a module is free or...

we make V a k[t]-module. The structure theorem then says V is a direct sum of cyclic modules, each of which is isomorphic to the module of the form k [...

\mathbf {P} (R)} the set of isomorphism classes of finitely generated projective modules over R; let also P n ( R ) {\displaystyle \mathbf {P} _{n}(R)} subsets...

between modules is an invertible module homomorphism. Jacobson Jacobson's density theorem Kähler differentials Kähler differentials Kaplansky Kaplansky's theorem...

(analytic number theory) Kaplansky's theorem on quadratic forms (number theory) Khinchin's theorem (probability) Kronecker's theorem (Diophantine approximation)...

the operation on pairs of module elements determines a group representation. The real utility of modules is that the kinds of modules that exist and...

-modules is Krull-Schmidt. Examples of semiperfect rings include: Left (right) perfect rings. Local rings. Kaplansky's theorem on projective modules Left...

research. Projective modules can be defined to be the direct summands of free modules. If R is local, any finitely generated projective module is actually...

Veblen–Young theorem. Von Neumann extended this fundamental result in projective geometry to the continuous dimensional case. This coordinatization theorem stimulated...

Noetherian ring. More generally the Krull dimension can be defined for modules over possibly non-commutative rings as the deviation of the poset of submodules...

subsets Countably generated module. (Kaplansky's theorem says that a projective module is a direct sum of countably generated modules.) This disambiguation...

homological machinery to these modules. Additionally, the theorem that every projective Z {\displaystyle \mathbb {Z} } -module is free generalizes in the...

from the original on 10 April 2008. Retrieved 29 May 2010. Goldfeld, Dorian (2003). "The Elementary Proof of the Prime Number Theorem: an Historical Perspective"...

terminology may persist: theorems often enough may still be referred to as conjectures, using the anachronistic names. Deligne's conjecture on 1-motives Goldbach's...

real numbers and its field of fractions M = R(x,y,z). The projective dimension of M as A-module is either 2 or 3, but it is independent of ZFC whether it...

2)-dimensional quadric in the (n − 1)-dimensional projective space. This is a basic construction in projective geometry. In this way one may visualize 3-dimensional...

Kaplansky's theorem (Kaplansky 1958), which states that a projective module is a direct sum of countably generated modules. More generally, a module over...

The Auslander–Buchsbaum formula relates depth and projective dimension. Theorem—Let M be a finite module over a noetherian local ring R. If pd R ⁡ M < ∞...

two non-zero modules are isomorphic, and all non-zero modules are standard. Connes (1976) and others proved that the following conditions on a von Neumann...

projective dimensions of all A-modules. Global dimension is an important technical notion in the dimension theory of Noetherian rings. By a theorem of...

sets with specific operations acting on their elements. Algebraic structures include groups, rings, fields, modules, vector spaces, lattices, and algebras...

Jacobson radical of a left-and-right Noetherian ring is precisely 0. Kaplansky's conjectures Köthe conjecture: if a ring has no nil ideal other than {...

ideal m/J. A deep theorem by Irving Kaplansky says that any projective module over a local ring is free, though the case where the module is finitely-generated...

possess the nice ideal and module theoretic properties of Dedekind domains, but usually only for finitely generated modules. Prüfer domains are named after...