I am looking for a precise definition of what is meant by an "algebra".
The reason is that I want to compare the difference between an "algebra" and other algebraic structures, i.e. a group, ring, field etc.
I have found the definitions of the others in books, while I haven't been able to find the definition of an algebra, surprisingly.
The only thing I found is in Folland's book on real analysis, p.21:
"An algebra of sets on some nonempty set $X$ is a nonempty collection $\mathcal{A}$ of subsets of $X$ that is closed under finite unions and complements..."
I hope someone can help.
There are several things called algebra in mathematics, including these:
Algebra over a field: a ring that is also a vector space.
Algebra of sets: a collection of sets closed under the usual set operations.
$\sigma$-algebra: a collection of sets closed under countably many set operations.
Algebra (or algebraic structure): the general object of study of universal algebra.