What is the best way to think about an R-algebra?

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I know that thinking a R-module is simply a ring R acting on a set (following the usual axioms), would it be safe to think of an R-algebra as the ring R acting on another ring?

This may seem convoluted, but I'm just having a little trouble getting through all the different definitions of an R-algebra.
 
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An R-Algebra is an R-module closed under some multiplication operation. i.e., an R-module that is also a ring.

For a special (associative) case, the set of n x n matrices with entries from a ring R is a dimension n^2 R-module closed under the usual matrix multiplication.
 

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