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Encyclopedia > Algebra over a commutative ring

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



  Results from FactBites:
 
Algebra (ring theory) - Wikipedia, the free encyclopedia (713 words)
In mathematics, specifically in ring theory, an algebra over a commutative ring is a generalization of the concept of an algebra over a field, where the base field K is replaced by a commutative ring R.
In this article, all rings and algebras are assumed to be unital and associative.
The free R-algebra on a set E is an algebra of polynomials with coefficients in R and noncommuting indeterminates taken from the set E.
Ring (mathematics) - Wikipedia, the free encyclopedia (1408 words)
A ring is a generalization of the integers, which itself is an example of a ring.
Rings that also satisfy commutativity for multiplication (such as the ring of integers) are called commutative rings.
Since any ring is both a left and right module over itself, it is possible to construct the tensor product of R over a ring S with another ring T to get another ring provided S is a central subring of R and T.
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