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:





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.


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