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...

kernel of a homomorphism is the relation describing how elements in the domain of the homomorphism become related in the image. A homomorphism is a function...

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

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...

{\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...

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...

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

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...

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

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

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...

(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...

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...

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...

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

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...

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...

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...

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

  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...

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...

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...

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...

/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...

which any n in N, m ↦ B(m, n) is an R-module homomorphism, and for any m in M, n ↦ B(m, n) is an S-module homomorphism. This satisfies B(r ⋅ m, n) = r ⋅ B(m...

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...

{\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...

{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 ) ⊗...

be thought of as a ring homomorphism from R into the ring of abelian group endomorphisms of M. The image of this homomorphism is a semiprimitive ring...

\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...