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.

Advertisements

About mathtuition88

http://mathtuition88.com
This entry was posted in math and tagged . Bookmark the permalink.

Leave a Reply

Fill in your details below or click an icon to log in:

WordPress.com Logo

You are commenting using your WordPress.com account. Log Out / Change )

Twitter picture

You are commenting using your Twitter account. Log Out / Change )

Facebook photo

You are commenting using your Facebook account. Log Out / Change )

Google+ photo

You are commenting using your Google+ account. Log Out / Change )

Connecting to %s