Algebra (ring theory) Guide, Meaning , Facts, Information and Description

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

Let R be a commutative ring and let S be another ring (not necessarily commutative). S is called an R-algebra if there is a ring homomorphism from R into the center of S. If S itself is commutative 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

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