mathematics, particularly in algebra, the injective hull (or injective envelope) of a module is both the smallest injective module containing it and the largest...

the field of category theory, the concept of injective object is a generalization of the concept of injective module. This concept is important in cohomology...

measure how far from injective a module is in terms of the injective dimension and represent modules in the derived category. Injective hulls are maximal essential...

injective metric space into which M can be embedded. In some sense it consists of all points "between" the points of M, analogous to the convex hull of...

injective hull of M. The injective hull is necessarily an injective module, and is unique up to isomorphism. The injective hull is also minimal in the sense...

complex analysis Injective hull, of a module Linear hull, another name for the linear span Skolem hull, of mathematical logic Hull, the common name of...

Ẽ(M) = E(M). The rational hull is readily identified within the injective hull. Let S=EndR(E(M)) be the endomorphism ring of the injective hull. Then an element...

essential subgroup. This divisible group D is the injective envelope of A, and this concept is the injective hull in the category of abelian groups. An abelian...

Noetherian complete local ring with residue field k, and choose E to be an injective hull of k (sometimes called a Matlis module). The dual DR(M) of a module...

essential submodule of the codomain. An injective hull of an object A is an essential monomorphism from A to an injective object. Hashimoto, Mitsuyasu (November...

conditions: R is Noetherian on one side and self-injective on one side. R is Artinian on a side and self-injective on a side. All right (or all left) R modules...

complete lattices form the injective objects for order-embeddings, and the Dedekind–MacNeille completion of S is the injective hull of S. Several researchers...

module. If R is an Artinian local ring then the Matlis module of R (the injective hull of the residue field) is the dualizing module. The Artinian local ring...

E(R/{\mathfrak {p}})\,} where E ( − ) {\displaystyle E(-)\,} denotes the injective hull and p {\displaystyle {\mathfrak {p}}\,} ranges over the prime ideals...

orthogonal convex hull is a special case of a much more general construction, the hyperconvex hull, which can be thought of as the smallest injective metric space...

module Projective cover Swan's theorem Quillen–Suslin theorem Injective module Injective hull Flat module Flat cover Coherent module Finitely-generated module...

Eckmann–Hilton duality Eckmann–Hilton argument Eckmann–Shapiro lemma Injective hull Awards Albert Einstein Medal (2008) Scientific career Fields Mathematics...

ideal m and residue field k = R/m. Let E(k) be a Matlis module, an injective hull of k, and let Ω be the completion of its dualizing module. Then for...

dimension n, the following are equivalent: R has finite injective dimension as an R-module; R has injective dimension n as an R-module; The Ext group Ext R i...

Lie algebra decomposes into the direct sum of its socle and cosocle.) Injective hull Radical of a module Cosocle Robinson 1996, p.87. J. L. Alperin; Rowen...

exactly when N is an essential submodule of M. In particular, M and its injective hull E(M) always have the same uniform dimension. It is also true that u...

envelope or injective hull of a module is a smallest injective module containing it. 3.  An injective resolution is a resolution by injective modules. 4...

nonsingularity has a strong interaction with right self injective rings as well. Theorem: If R is a right self injective ring, then the following conditions on R are...

called injective hull) is a maximal essential extension, or a minimal embedding in an injective module. 3.  An injective cogenerator is an injective module...

categories. Every Grothendieck category contains an injective cogenerator. For example, an injective cogenerator of the category of abelian groups is the...

i=\dim M.} If R is complete and d its Krull dimension and if E is the injective hull of k, then Hom R ⁡ ( H m d ( − ) , E ) {\displaystyle \operatorname...

x) and an inflation x→I(x) (the injective hull of x) such that both P(x) and I(x) are in the category of pro/injective objects. A Frobenius category E...

to (non-unique) isomorphism. Flat covers are in some sense dual to injective hulls, and are related to projective covers and torsion-free covers. The...

category A {\displaystyle {\mathcal {A}}} is moreover cocomplete and has injective hulls (e.g. if it is a Grothendieck category), then a Serre subcategory C...

over the hull. The current is strong and often changeable, gradually opening up holes in the ship's hull. Salt corrosion eats away at the hull, and it...