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# 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 $Z(A)$.

## Mathematics Subject Classification

20C99*no label found*16S99

*no label found*13B02

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## Comments

## for the people, by the people

While this entry may be technically correct, I still woulden't know what algebra was by reading it ;) May I suggest a short history of the invention of algebra, as well something to to tune of "a system to manipulate mathematical formulas"

## Re: for the people, by the people

yep.. we had a talk on this.

point is algebra has a very precise tehcnical meaning.

However.. being this one of the site values (user centred), feel free to add the entry "algebra" with "branch" type.

and also.. we're working on making a better browsable site

f

G -----> H G

p \ /_ ----- ~ f(G)

\ / f ker f

G/ker f

## Re: for the people, by the people

"topic" type, you mean =)

-apk

## algebra

In your definition of an algebra, there are many authors, who do not require that the ring has an identity at all. The Bourbaki school does, but as mentioned many mathematicians do not. The presence of an identity element helps, when maximal ideals (or one sided ideals are needed of course.)