# algebras

Let $K$ be a commutative   unital ring (often a field) and $A$ a $K$-module. Given a bilinear mapping $b:A\times A\rightarrow A$, we say $(K,A,b)$ is a $K$-algebra   . We usually write only $A$ for the tuple $(K,A,b)$.

This definition is a compact method to encode the property that our multiplication is distributive: the multiplication is additive in both variables translates to

 $(a+b)c=ac+bc,\qquad a(b+c)=ab+ac\qquad a,b,c\in A.$

Furthermore, the assumption that scalars can be passed in and out of the bilinear product  translates to

 $(la)b=l(ab)=a(lb),\qquad a,b\in A,l\in K.$
###### Proposition 2.

Given an algebra $A$, the set

 $Z_{0}(A)=\{z\in A:za=az,a\in A\}.$

$Z_{0}(A)$

###### Proof.

For now let elements of $A$ be denoted with $\hat{a}$ to distinguish them from scalars. As a module $0\hat{a}=\hat{0}$ for all $a\in A$. Then

 $\hat{0}\hat{a}=(0\hat{a})\hat{a}=(\hat{a})(0\hat{a})=\hat{a}\hat{0}.$

So $\hat{0}\in Z_{0}(A)$.

Also given $\hat{z},\hat{w}\in Z_{0}(A)$ then for all $a\in A$,

 $(\hat{z}+\hat{w})\hat{a}=\hat{z}\hat{a}+\hat{w}\hat{a}=\hat{a}\hat{z}+\hat{a}% \hat{w}=\hat{a}(\hat{z}+\hat{w}).$

So $\hat{z}+\hat{w}\in A$.

Finally, given $l\in K$ we have

 $(l\hat{z})\hat{a}=l(\hat{z}\hat{a})=l(\hat{a}\hat{z})=\hat{a}(l\hat{z}).$

Although this set $Z(A)$ appears like a reasonable object to define as the center of an algebra, it is usually preferable to produce a subalgebra, not simply a submodule, and for this we need elements that can be regrouped in products associatively, that is, that lie in the nucleus. So the center is commonly defined as

 $Z(A)=\{z\in A:za=az,z(ab)=(za)b,a(zb)=(az)b,(ab)z=a(bz),a,b\in A\}.$

When the algebra $A$ has an identity     (unity) $1$ then we can go further to identify $K$ as a subalgebra of $A$ by $l1$. Then we see this subalgebra is necessarily in the center of $A$. As a converse, given a unital ring $R$ (associativity is necessary), the center of the ring forms a commutative unital subring over which $R$ is an algebra. In this way unital rings and associative unital algebras are often interchanged.

Title algebras Algebras 2013-03-22 16:27:20 2013-03-22 16:27:20 Algeboy (12884) Algeboy (12884) 7 Algeboy (12884) Definition msc 17A01 ring NonAssociativeAlgebra FreeAssociativeAlgebra TopicEntryOnTheAlgebraicFoundationsOfMathematics algebra