algebra, a module homomorphism is a function between modules that preserves the module structures. Explicitly, if M and N are left modules over a ring...

any homomorphism of mathematical objects, is just a mapping that preserves the structure of the objects. Another name for a homomorphism of R-modules is...

map defined by the matrix. The kernel of a homomorphism is reduced to 0 (or 1) if and only if the homomorphism is injective, that is if the inverse image...

{\displaystyle f:N\to M} of graded modules, called a graded morphism or graded homomorphism , is a homomorphism of the underlying modules that respects grading; i...

scalar multiplication. A module homomorphism, also called a linear map between modules, is defined similarly. An algebra homomorphism is a map that preserves...

of some other module, then it is already a direct summand of that module; also, given a submodule of a module Y, any module homomorphism from this submodule...

characterization of a faithfully flat homomorphism for a not-necessarily-flat homomorphism. Given an injective local homomorphism ( R , m ) ↪ ( S , n ) {\displaystyle...

This article contains special characters. Without proper rendering support, you may see question marks, boxes, or other symbols. 0 (zero) is a number representing...

  An endomorphism is a module homomorphism from a module to itself. 2.  The endomorphism ring is the set of all module homomorphisms with addition as addition...

homomorphism g : P → M, there exists a module homomorphism h : P → N such that f h = g. (We don't require the lifting homomorphism h to be unique; this is not a...

f:R\to S} is a ring homomorphism and if M is a right S-module and N a left S-module, then there is the canonical surjective homomorphism: M ⊗ R N → M ⊗ S...

may be different.) On the other hand, a module homomorphism M → K is a pure embedding if the induced homomorphism between the tensor products C ⊗ M → C...

(left or right) modules over the same ring, and let f : M → N be a module homomorphism. If M is simple, then f is either the zero homomorphism or injective...

definition are also used for the more general case of modules over a ring; see Module homomorphism. If a linear map is a bijection then it is called a linear...

of two objects between which a homomorphism is given, and of the kernel and image of the homomorphism. The homomorphism theorem is used to prove the isomorphism...

a + B is called the quotient map or the projection map, and is a module homomorphism. The addition operation on A/B is defined for two equivalence classes...

highest weight λ. We know from the section about homomorphisms of Verma modules that there exists a homomorphism W w ′ ⋅ λ → W w ⋅ λ {\displaystyle W_{w'\cdot...

R\to T} is a ring homomorphism that maps every element of S to a unit (invertible element) in T, there exists a unique ring homomorphism g : S − 1 R → T...

be viewed as a right R-module homomorphism r ↦ fr so that ffr = r, or f can also be viewed as a left R-module homomorphism r ↦ rf, where rff = r. This process...

relationship among quotients, homomorphisms, and subobjects. Versions of the theorems exist for groups, rings, vector spaces, modules, Lie algebras, and other...

left modules over R is the category whose objects are all left modules over R and whose morphisms are all module homomorphisms between left R-modules. For...

kernels are a generalization of the kernels of group homomorphisms, the kernels of module homomorphisms and certain other kernels from algebra. Intuitively...

/2\mathbf {Z} } The first homomorphism maps each element i in the set of integers Z to the element 2i in Z. The second homomorphism maps each element i in...

\mathbb {Z} } -module homomorphism. Since an abelian group is a Z {\displaystyle \mathbb {Z} } -module, it may be defined as a group homomorphism between abelian...

{\displaystyle f:E\to N} from a set E to a left R-module N, there exists a unique module homomorphism f ¯ : R ( E ) → N {\displaystyle {\overline {f}}:R^{(E)}\to...

are unital, then a homomorphism satisfying f(1A) = 1B is said to be a unital homomorphism. The space of all K-algebra homomorphisms between A and B is...

module has a canonical homomorphism to the dual of its dual (called the double dual). A reflexive module is one for which the canonical homomorphism is...

{O}}(U_{\alpha })} -module. For each pair of open affine subschemes V ⊆ U {\displaystyle V\subseteq U} of X {\displaystyle X} , the natural homomorphism O ( V ) ⊗...

bilinear form can be extended to include modules over a ring, with linear maps replaced by module homomorphisms. When K is the field of complex numbers...

dropped. A ring homomorphism f is said to be an isomorphism if there exists an inverse homomorphism to f (that is, a ring homomorphism that is an inverse...