Is localization faithfully flat?
A flat local ring homomorphism of local rings is faithfully flat. Flatness meshes well with localization. Lemma 10.39.
Is flatness a local property?
Flatness is a local property: M is flat over R iff Mp is flat over Rp for all prime ideals p iff Mm is flat over Rm for all maximal ideals m. (4): We deduce that if M is flat over R then S−1M is flat over S−1R. It remains to check that if Mm is flat over Rm for all maximal ideals m then M is in fact flat over R.
What is a faithful module?
A faithful module M is one where the action of each r ≠ 0 in R on M is nontrivial (i.e. r ⋅ x ≠ 0 for some x in M). Equivalently, the annihilator of M is the zero ideal.
Are projective modules flat?
flat modules. Every projective module is flat. The converse is in general not true: the abelian group Q is a Z-module which is flat, but not projective. Conversely, a finitely related flat module is projective.
What is regular ring?
In commutative algebra, a regular ring is a commutative Noetherian ring, such that the localization at every prime ideal is a regular local ring: that is, every such localization has the property that the minimal number of generators of its maximal ideal is equal to its Krull dimension.
Are free modules flat?
In particular, a direct limit of free modules is flat. Conversely, every flat module can be written as a direct limit of finitely-generated free modules. Direct products of flat modules need not in general be flat.
Is being finitely generated a local property?
The property of being finitely generated is local.
How do you prove a module is flat?
A module is faithfully flat if taking the tensor product with a sequence produces an exact sequence if and only if the original sequence is exact. Although the concept is defined for modules over a non-necessary commutative ring, it is used mainly for commutative algebras.
What is module number?
Module(absolute value) of a positive number or zero is the number itself and module of a negative number is called its contrary number i.e. |a| = a if a ≥ = 0 and. |a| = -a if a < 0. From the definition is clear that the absolute value of every rational number different from zero is a positive number.
Is a free module flat?
Flatness is related to various other module properties, such as being free, projective, or torsion-free. In particular, every flat module is torsion-free, every projective module is flat, and every free module is projective. There are finitely generated modules that are flat and not projective.
What is direct Summand?
Given the direct sum of additive Abelian groups , and are called direct summands. The map defined by is called the injection of the first summand, and the map defined by is called the projection onto the first summand. Similar maps are defined for the second summand .
Are fields local rings?
Examples. All fields (and skew fields) are local rings, since {0} is the only maximal ideal in these rings.
Are submodules of free modules free?
Submodules of free modules every submodule of a free R-module is itself free; every ideal in R is a free R-module; R is a principal ideal domain.
What is the rank of a module?
The rank of a free module M over an arbitrary ring R( cf. Free module) is defined as the number of its free generators. For rings that can be imbedded into skew-fields this definition coincides with that in 1). In general, the rank of a free module is not uniquely defined.
What are local properties?
In mathematics, a mathematical object is said to satisfy a property locally, if the property is satisfied on some limited, immediate portions of the object (e.g., on some sufficiently small or arbitrarily small neighborhoods of points).
What do you mean by finitely generated?
In algebra, a finitely generated group is a group G that has some finite generating set S so that every element of G can be written as the combination (under the group operation) of finitely many elements of the finite set S and of inverses of such elements.
What is the purpose of homological algebra?
Homological algebra affords the means to extract information contained in these complexes and present it in the form of homological invariants of rings, modules, topological spaces, and other ‘tangible’ mathematical objects. A powerful tool for doing this is provided by spectral sequences.
Is Z an Injective module?
The rationals Q (with addition) form an injective abelian group (i.e. an injective Z-module). The factor group Q/Z and the circle group are also injective Z-modules. The factor group Z/nZ for n > 1 is injective as a Z/nZ-module, but not injective as an abelian group.