In this definition, all rings are assumed to be rings with identity and all ring homomorphismsMathworldPlanetmath are assumed to be unital.

Let R be a ring. An algebra over R is a ring A together with a ring homomorphism f:RZ(A), where Z(A) denotes the center of A. A subalgebra of A is a subset of A which is an algebra.

Equivalently, an algebra over a ring R is an R–module A which is a ring and satisfies the property


for all rR and all x,yA. Here denotes R-module multiplication and * denotes ring multiplication in A. One passes between the two definitions as follows: given any ring homomorphism f:RZ(A), the scalar multiplication rule


makes A into an R-module in the sense of the second definition. Conversely, if A satisfies the requirements of the second definition, then the function f:RA defined by f(r):=r1 is a ring homomorphism from R into Z(A).

Title algebra
Canonical name Algebra
Date of creation 2013-03-22 11:48:37
Last modified on 2013-03-22 11:48:37
Owner djao (24)
Last modified by djao (24)
Numerical id 17
Author djao (24)
Entry type Definition
Classification msc 20C99
Classification msc 16S99
Classification msc 13B02
Defines subalgebra