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...

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

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...

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

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...

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...

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

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

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

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...

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...

\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...

{\displaystyle L} . In this way, projective space acquires a universal property. The universal way to determine a map to projective space is to map to the projectivization...

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...

Differential module Five lemma Short five lemma Snake lemma Nine lemma Extension (algebra) Central extension Splitting lemma Projective module Injective module Projective...

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...

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...

one projective module over R. For example, this includes fractional ideals of algebraic number fields, since these are rank one projective modules over...

particular free and projective modules, are torsion-free, but the converse need not be true. An example of a torsion-free module that is not flat is the...

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...

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...

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

{\displaystyle R^{n}} for a positive integer n. Since every free module is projective module, then exists right inverse of the projection map (it suffices...

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...

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...

theory. They introduced fundamental concepts, including those of projective module, weak dimension, and what is now called the Cartan–Eilenberg resolution...

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...