What is the difference between a Ring and an Algebra?

In mathematics, I want to know what is indeed the difference between a ring and an algebra?

Answer

A ring R has operations + and × satisfying certain axioms which I won’t repeat here. An (associative) algebra A similarly has operations + and × satisfying the same axioms (it doesn’t need a multiplicative identity, but this axiom isn’t always assumed in rings either), plus an additional operation :R×AA, where R is some ring (often a field) that satisfies some axioms making it compatible with the multiplication and addition in A. You should think of this as an analogue of scalar multiplication in vector spaces.

Note also that there are non-associative algebras, so the axioms on multiplication can be weakened from those in rings.

As a vague summary, the algebraic structure of a ring is entirely internal, but in an algebra there is also structure coming from interaction with an external ring of scalars.

Attribution
Source : Link , Question Author : user70795 , Answer Author : Mike Pierce

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