In fact a ring S is an R algebra, relative to a base ring R, just as S might be an R module. The base ring R should be stated if it is not implied by context.
The ring S is a left R algebra if S is a left unitary R module, and the action of R can be interchanged with multiplication in S. For every a in R and every x and y in S:
a(xy) = (ax)y =x(ay)
This definition is easier to understand if we assume all rings contain 1, and homomorphisms map 1 to 1. (This is the default assumption throughout my website.) Set x = 1 and ax becomes an element of S; we may as well call it f(a). Since S is an R module, f respects addition in R. How about multiplication? In the following, a and b are elements of R, and x lies in S.
f(ab) = (ab)1 = a(b1) = af(b) = a(1f(b)) = (a1)f(b) = f(a)f(b)
1*1 = 1 (S is a unitary module)
f(a)x = ax1 = xa1 = xf(a)
The function f respects multiplication, as well as addition, and it carries 1 to 1. The image of R in S defines an R homomorphism into S. Furthermore, since f(a) and x commute, R maps into the center of S.
Conversely, let f(R) be a ring homomorphism into the center of S. This makes S an R module, and since 1 maps to 1, S is a unitary module. Use the properties of a ring homomorphism, and the fact that the image lies in the center of S, to prove a(xy) = (ax)y =x(ay), and you have completed the circle. The ring S is an R algebra iff a ring homomorphism carries R into the center of S.
The image of ab in S is f(a)f(b), which is the same as f(b)f(a), or the image of ba. Any noncommutative aspects of R are left on the cuttingroom floor. It is equivalent to think of S as a right R algebra, so we may as well call it an R algebra and be done with it.
Technically, the homomorphism is part of the algebra. It is the action of R on S. Sometimes R is replaced with the image of R in S, so that R embeds in S. Now the homomorphism becomes the identity map.
If S is a division ring then S is a division algebra.
If S is a flat R module then it is a flat R algebra.
If S is a finitely generated R module then S is a finite R algebra. However, S is a finitely generated algebra, or an algebra of finite type, if it is finitely generated over R as a ring. Adjoin finitely many elements to R, apply all ring operations, and mod out by certain relations.
If the homomorphic image of R is commutative, it forms an R module and a finite R algebra. Thus Zn is a Z algebra. Trivially, R (commutative) is an R algebra.
Every ring is a Z algebra. Map 1 to 1, and Z maps into the center of S.
Polynomials, power series, and laurent series, in multiple indeterminants, that may or may not commute with each other, are examples of R algebras.
The complex numbers and the quaternions are real algebras.
If S is an R algebra, then the ring of n×n matrices over S is an R algebra. Map R into diagonal matrices, with 1 leading to the identity matrix. The polynomials over S also form an R algebra, and so on.