## Opposite Algebra

Let $(A,+,\mu)$ be an $R$-algebra. Define an operation $\mu': A\times A\to A$ by $\mu'(a,b)=\mu(b,a)=ba$. This algebra $(A,+,\mu')$ is called the opposite algebra to $A$. We verify that it forms an $R$-algebra.

Bilinearity comes from the following computations:

$\mu'(a_1+a_2,b)=b(a_1+a_2)=ba_1+ba_2=\mu'(a_1,b)+\mu'(a_2,b)$

$\mu'(a,b_1+b_2)=(b_1+b_2)a=b_1a+b_2a=\mu'(a,b_1)+\mu'(a,b_2)$

$\mu'(ar,b)=bar=\mu'(a,b)r$

$\mu'(a,br)=bra=bar=\mu'(a,b)r$

Associativity is true from $\mu'(\mu'(a,b),c)=cba=\mu'(a,\mu'(b,c))$

The unity element is the same unity element $1_A$: $\mu'(x,1_A)=1_Ax=x$, $\mu'(1_A,x)=x1_A=x$.