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

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...

Then, for a homomorphism f : G → H, (f(u),f(v)) is an arc (directed edge) of H whenever (u,v) is an arc of G. There is an injective homomorphism from G to...

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)...

mathematics, a ring homomorphism is a structure-preserving function between two rings. More explicitly, if R and S are rings, then a ring homomorphism is a function...

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...

of Heinz Hopf (1935). Whitehead's original homomorphism is defined geometrically, and gives a homomorphism J : π r ( S O ( q ) ) → π r + q ( S q ) {\displaystyle...

Monoid homomorphisms are sometimes simply called monoid morphisms. Not every semigroup homomorphism between monoids is a monoid homomorphism, since it...

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

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...

antihomomorphism is the same thing as a homomorphism. The composition of two antihomomorphisms is always a homomorphism, since reversing the order twice preserves...

\!\!+\;m)&&=\ h\ l\oplus h\ m.\end{aligned}}} The homomorphism lemma states that h is a homomorphism if and only if there exists an operator ⊕ {\displaystyle...

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

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

f. A semigroup homomorphism between monoids preserves identity if it is a monoid homomorphism. But there are semigroup homomorphisms that are not monoid...

functional analysis, a topological homomorphism or simply homomorphism (if no confusion will arise) is the analog of homomorphisms for the category of topological...

exists a group homomorphism h ∗ : π n ( X ) → H n ( X ) , {\displaystyle h_{*}\colon \pi _{n}(X)\to H_{n}(X),} called the Hurewicz homomorphism, from the n-th...

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

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...

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

for any group homomorphism f : G → H {\displaystyle f:G\to H} . Note that f op {\displaystyle f^{\text{op}}} is indeed a group homomorphism from G op {\displaystyle...

continuous homomorphism between real Lie groups turns out to be (real) analytic. The composition of two Lie homomorphisms is again a homomorphism, and the...

transfer defines, given a group G and a subgroup H of finite index, a group homomorphism from G to the abelianization of H. It can be used in conjunction with...

ideals as the main references. The three isomorphism theorems, called homomorphism theorem, and two laws of isomorphism when applied to groups, appear explicitly...

H^{*}(M;\mathbb {C} )} , called the Chern–Weil homomorphism, where on the right cohomology is de Rham cohomology. This homomorphism is obtained by taking invariant polynomials...

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...

for any finite extension K of k the ring R ⊗k K is a regular ring. A homomorphism of rings from R to S is called regular if it is flat and for every p ∈ Spec(R)...

homological algebra, the Bockstein homomorphism, introduced by Meyer Bockstein (1942, 1943, 1958), is a connecting homomorphism associated with a short exact...

are R-S-bimodules, then a map f : M → N is a bimodule homomorphism if it is both a homomorphism of left R-modules and of right S-modules. An R-S-bimodule...