algebra

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

Let $R$ be a ring. An algebra over $R$ is a ring $A$ together with a ring homomorphism $f\colon R\to Z(A)$ , 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

r\cdot(x*y)=(r\cdot x)*y=x*(r\cdot y)

for all $r\in R$ and all $x,y\in A$ . Here $\cdot$ denotes $R$ -module multiplication and $*$ denotes ring multiplication in $A$ . One passes between the two definitions as follows: given any ring homomorphism $f\colon R\longrightarrow Z(A)$ , the scalar multiplication rule

r\cdot b:=f(r)*b

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 $f\colon R\to A$ defined by $f(r):=r\cdot 1$ 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