Algebra (ring theory)

In ring theory, an algebra over a base ring is a generalization of the concept of associative algebra.

Let R be a commutative ring. An R-algebra is a ring S together with a ring homomorphism from R to the center of S. If S itself is commutative then it is called a commutative R-algebra.

The notion of R-algebra generalizes that of an associative algebra: if K is a field, then any associative algebra over K is a K-algebra and vice-versa. Every R-algebra is also an R-module in an obvious manner.

Examples

• Any ring S can be considered as an algebra over its center R.
• Any ring S can be considered as a Z-algebra in a unique way.
• Every polynomial ring R[x₁, ..., x_n] is a commutative R-algebra.

See also

• algebra over a field (not necessarily associative)
• commutative algebra