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In our algebra course,several times we are introduced to the concept of an algebra.An algebra over a field $F$ is roughly a ring and an $F$-vector space.An algebra homomorphism is basically a ring homomorphism and a linear transformation.But then in our module theory course,our instructor introduced the term 'algebra' a bit differently.He told us that a commutative ring $A$ together with a ring homomorphism $f:R\to A$ (Where $R$ is a commutative ring) is called an $R$-algebra.Then he showed that if there exists such an $f$ ,then $A$ is an $R$-module.But these two definitions do not seem to match.I don't think they are equivalent.Then what is the connection between these two things?

Kishalay Sarkar
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  • How my professor described it, (in a very elementary way) is a vector space with defined vector multiplication. The idea can easily transfer to modules. Think of sets of matrices of the same size. If they are not square - can't multiply; if they are square- can multiply. – David P Sep 07 '22 at 03:49

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A $K$-algebra $A$ is a vector space over a field $K$ equipped with a bilinear product, see here. If it is not necessarily associative or commutative it is often called a non-associative algebra.

For a commutative ring $R$, an $R$-algebra $A$ is, as you said, is a ring with identity together with a ring homomorphism $f\colon R \to A$ such that the subring $f(R)$ of $A$ is contained within the center of $A$."

Reference: What exactly is an $R$-algebra?

For the difference or similarities of the two notions see some posts here:

Definition of R-Algebra

Difference between an R algebra and an algebra as described by Rudin

The first definition includes Lie algebras, Jordan algebras, alternative algebras, flexible algebras, Jacobi-Jordan algebras and many more.

Dietrich Burde
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