from free to projective modules: a module P is projective if and only if for every surjective module homomorphism f : N ↠ M and every module homomorphism...

R. These are the modules that behave very much like vector spaces. Projective Projective modules are direct summands of free modules and share many of...

algebra, flat modules include free modules, projective modules, and, over a principal ideal domain, torsion free modules. Formally, a module M over a ring...

Kaplansky's theorem states a projective module over a (possibly non-commutative) local ring is free. Sometimes, whether a module is free or not is undecidable...

equivalent conditions on a module. Some crossover occurs for projective or flat modules. A finitely generated projective module is finitely presented, and...

homomorphism from all of Y to Q. This concept is dual to that of projective modules. Injective modules were introduced in (Baer 1940) and are discussed in some...

dimension zero if and only if it is a projective module. If M does not admit a finite projective resolution then the projective dimension is infinite. For example...

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

In category theory, the notion of a projective object generalizes the notion of a projective module. Projective objects in abelian categories are used...

a direct summand of free modules. In particular, every free module is projective. 2.  The projective dimension of a module is the minimal length of (if...

commutative algebra concerning the relationship between free modules and projective modules over polynomial rings. In the geometric setting it is a statement...

the algebraic concept of projective modules and gives rise to a common intuition throughout mathematics: "projective modules over commutative rings are...

and sufficient for OL to be a projective module over Z[G]. It is certainly therefore necessary for it to be a free module. It leaves the question of the...

concept of a projective space originated from the visual effect of perspective, where parallel lines seem to meet at infinity. A projective space may thus...

Look up projection, projecting, projective, or projector in Wiktionary, the free dictionary. Projection, projections or projective may refer to: Projection...

f − g factors through a projective module. The stable module category is defined by setting the objects to be the R-modules, and the morphisms are the...

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...

submodule of a projective module over it is projective. Similarly, an integral domain is a Dedekind domain if and only if every divisible module over it is...

category theory, a projective cover of an object X is in a sense the best approximation of X by a projective object P. Projective covers are the dual...

f(A)\oplus s(C).} In particular, any module over a semisimple ring is injective and projective. Since "projective" implies "flat", a semisimple ring is...

general modules, and become isomorphisms if the modules E and F are restricted to being finitely generated projective modules (in particular free modules of...

\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...

Dedekind domain, a finitely generated torsion-free module is projective and a projective module is reflexive (the existence of a dual basis). Bourbaki 1998...

The Apollo command and service module (CSM) was one of two principal components of the United States Apollo spacecraft, used for the Apollo program, which...

Pure injective and pure projective modules follow closely from the ideas of Prüfer's 1923 paper. While pure projective modules have not found as many applications...

especially its simple modules, projective modules, and indecomposable modules. A (left) principal indecomposable module of a ring R is a (left) submodule...

length k ≤ n. This upper bound on the projective dimension is sharp, that is, there are modules of projective dimension exactly n. The standard example...

that a projective module is a direct sum of countably generated modules. More generally, a module over a possibly non-commutative ring is projective if and...

variety Projective linear group Projective module Projective line Projective object Projective transformation Projective hierarchy Projective connection...

as non-isomorphic projective indecomposables have non-isomorphic socles. The multiplicity of a projective indecomposable module as a summand of the...