composition of ring homomorphisms S → T and R → S is a ring homomorphism R → T. For each ring R, the identity map R → R is a ring homomorphism. Therefore...

semigroup homomorphism between groups is necessarily a group homomorphism. A ring homomorphism is a map between rings that preserves the ring addition...

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...

is 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...

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...

+ I is a surjective ring homomorphism, sometimes called the natural quotient map or the canonical homomorphism. The quotient ring R / {0} is naturally...

matrix ring. A homomorphism between two R-algebras is an R-linear ring homomorphism. Explicitly, φ : A1 → A2 is an associative algebra homomorphism if φ...

presented). A ring homomorphism R → S {\displaystyle R\to S} is flat if S is a flat R-module for the module structure induced by the homomorphism. For example...

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)...

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