Algebra Homomorphisms
A homomorphism between two associative R-algebras is an R-linear ring homomorphism. Explicitly, is an associative algebra homomorphism if
For a homomorphism of unital associative R-algebras, we also demand that
The class of all unital associative R-algebras together with algebra homomorphisms between them form a category, sometimes denoted R-Alg.
The subcategory of commutative R-algebras can be characterized as the coslice category R/CRing where CRing is the category of commutative rings.
Read more about this topic: Associative Algebra
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—José Ortega Y Gasset (1883–1955)