Algebra (ring theory)
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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, ..., xn] is a commutative R-algebra.
See also
- algebra over a field (not necessarily associative)
- commutative algebra