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I saw the definition of algebra over a ring and algebra over a field but I am not sure how to interpret it.

Many references would directly mention let $A$ be a $k$-algebra or $A$ be an $R$-algebra but what exactly it mean?

Obviously if I go through the definition it would mean the existence of a homomorphism $\rho \colon k \to R$ such that $\rho(k) \subset Z(R)$ but it seems too much to keeping defining homomorphisms and remember the property they satisfy. Is there any other equivalent definition which will let the grasp the concept immediately?

Similarly how about an $R$-algebra?

Jendrik Stelzner
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permutation_matrix
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  • An $,R$-algebra is simply a ring $,A,$ containing a central image of $,R,,$ i.e. $,A,$ contains a subring $,R' \cong R/I,$ such that the elements of $,R'$ commute with all elements of $,A,, $ e.g. rings of polynomials, power series and matrices over a ("coefficient") ring $A.,$ See here for some motivation via universal properties. – Bill Dubuque Dec 16 '22 at 16:43
  • See this post for $R$-algebras and $K$-algebras, and the linked posts. – Dietrich Burde Dec 16 '22 at 17:10
  • For a more general perspective on the module viewpoint see here. – Bill Dubuque Dec 16 '22 at 20:43
  • @MarianoSuárez-Álvarez, and an algebra over a field is a vector space right? it contains an homomorphic image of a field so it is injective. But somewhere I saw it is a vector space with a bilinear form. Where does this billinearity come from If I am not mistaken? – permutation_matrix Dec 28 '22 at 16:15
  • @MarianoSuárez-Álvarez, it is linear map when one of the coordinate in $A \times A$ is fixed – permutation_matrix Dec 29 '22 at 03:45
  • @MarianoSuárez-Álvarez, I read about it on the wikipedia page of Algebra over a field here https://en.m.wikipedia.org/wiki/Algebra_over_a_field – permutation_matrix Dec 29 '22 at 05:56

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