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?

Solution 1:

A ring $R$ has operations $+$ and $\times$ satisfying certain axioms which I won't repeat here. An (associative) algebra $A$ similarly has operations $+$ and $\times$ 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 $\cdot\;\colon R\times A\to A$, 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.

Solution 2:

One thing that complicates answering this question is that rings are almost always assumed to be associative, but algebras are frequently not assumed to be associative. (In other words, my impression is that it's more common to allow 'algebra' to name something nonassociative than it is to use 'ring' to mean something nonassociative.)

Nonassociative algebras are not rare: Lie algebras and Jordan algebras are common nonassociative algebras.

Associative algebras are not rare either: Every single ring $R$ is an associative algebra over its center!

Both structures might or might not be defined to have an identity, so we'll just overlook that feature.

Here's my take, (even though I think Matt Pressland's answer is pretty good already.) $R$ is a commutative ring.

An associative $R$-algebra $A$ is certainly a ring, and a nonassociative algebra may still be counted as a nonassociative ring.

The extra ingredient is an $R$ module structure on $A$ which plays well with the multiplication in $A$. (This was well described before by Matt P: indeed, they are like "scalars".)

In a nutshell, that module action and compatilibity is described by a ring homomorphism from $R$ into the center of $End(A)$, the ring of additive endomorphisms of $A$.

Solution 3:

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 $k$.