Algebra ring theory - Formal definition

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Algebra ring theory - Formal definition
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Algebra ring theory, Algebra ring theory - Algebra homomorphisms, Algebra ring theory - Constructions, Algebra ring theory - Examples, Algebra ring theory - Formal definition, algebra over a field, associative algebra, commutative algebra

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Algebra ring theory - Formal definition: Encyclopedia II - Algebra ring theory - Formal definition

Let R be a commutative ring. An R-algebra is a set A which has the structure of both a ring and an R-module in such a way that ring multiplication is an R-bilinear map. Explicity, we must have If A itself is commutative (as a ring) then it is called a commutative R-algebra. Starting with an R-module A, we get an R-algebra by equipping A with an R-bilinear map A × A< ...

An algebra homomorphism between two R-algebras is just an R-linear ring homomorphism. Explicity, is an algebra homomorphism if The class of all 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 ...

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