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In algebra, a normal homomorphism is a ring homomorphism R → S {\displaystyle R\to S} that is flat and is such that for every field extension L of the...

of this element. Thus a semigroup homomorphism between groups is necessarily a group homomorphism. A ring homomorphism is a map between rings that preserves...

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...

There is some group homomorphism G → H {\displaystyle G\to H} whose kernel is N {\displaystyle N} . There exists a group homomorphism ϕ : G → H {\displaystyle...

homomorphism sometimes means a map which respects not only the group structure (as above) but also the extra structure. For example, a homomorphism of...

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...

composition of module homomorphisms is again a module homomorphism, and the identity map on a module is a module homomorphism. Thus, all the (say left)...

the (usual) Gysin homomorphism induced by the zero-section embedding X ′ ↪ N {\displaystyle X'\hookrightarrow N} . The homomorphism i! encodes intersection...

covering map p : G → H is a continuous group homomorphism. The map p is called the covering homomorphism. A frequently occurring case is a double covering...

groups. Let G and H be groups, and let f : G → H be a homomorphism. Then: The kernel of f is a normal subgroup of G, The image of f is a subgroup of H, and...

{Gray}\longrightarrow }~\operatorname {coker} c} where d is a homomorphism, known as the connecting homomorphism. Furthermore, if the morphism f is a monomorphism...

In abstract algebra, a normal extension is an algebraic field extension L/K for which every irreducible polynomial over K that has a root in L splits into...

structure, too. In particular, a bounded-lattice homomorphism (usually called just "lattice homomorphism") f {\displaystyle f} between two bounded lattices...

groups, then a Lie group homomorphism f : G → H is a smooth group homomorphism. In the case of complex Lie groups, such a homomorphism is required to be a...

from their relation to homomorphisms. The first isomorphism theorem states that the image of any group G under a homomorphism is always isomorphic to...

homomorphism in one of these categories, and K is its kernel in the usual algebraic sense, then K is a subalgebra of X and the inclusion homomorphism...

advisable to specify which is the normal subgroup. If G = N ⋊ H {\displaystyle G=N\rtimes H} , then there is a group homomorphism φ : H → A u t ( N ) {\displaystyle...

field F can be viewed as left modules for the group algebra F[G]. Every homomorphism of left F[G]-modules ϕ : F [ G ] → K {\displaystyle \phi :F[G]\rightarrow...

λ(v). If f: R → S is a homomorphism between commutative rings, then S can be viewed as an algebra over R. The ring homomorphism f is then called the structure...

dimension of an algebraic variety. Field homomorphism A field homomorphism between two fields E and F is a ring homomorphism, i.e., a function f : E → F such...