TITLE: What is the difference between a Ring and an Algebra?
QUESTION [62 upvotes]: In mathematics, I want to know what is indeed the difference between a ring and an algebra?

REPLY [5 votes]: For a commutative ring $k$, a $k$-algebra is a ring $A$ together with an extra datum: a homomorphism from  $k$ into the center of $A$.
The definition allows non-associative algebras if you allow non-associative rings. The most well-understood case occurs when $k$ is a field, and the right way to think about the general case is as an algebraic family of algebras over fields---one for each prime ideal of $k$.