Abstract Nonsense

Crushing one theorem at a time

Opposite Ring and Antihomomorphism


Point of Post: In this post we discuss the notion of opposite rings and antihomomorphisms and prove some basic theorems regrading them.

\text{ }

Motivation

There is a natural way to take non-commutative rings and create new rings out of them. In particular, you take the ring and just pretend the multiplication has its order reversed.  In other words if we have a ring whose multiplication is denoted \ast then we can create a new multiplication called, say, \ast' given by a\ast' b=b\ast a. What seems to be true then is that while \text{id}_R:(R,+,\ast)\to (R,+,\ast') is not a homomorphism it is something that happens quite often a morphism which reverses order–the come up often when something takes “inverses”.

\text{ }

Opposite Rings and Antihomomorphisms

Let (R,+,\ast) be a ring, define the opposite ring denoted R^\text{op} to be the ring (R,+,\ast_\text{op}) where a\ast_\text{op}b=b\ast a. It’s easy to see that R^\text{op} is a ring.

\text{ }

An obvious question is as to whether R\cong R^\text{op}. “Usually” this is true, not in the sense that in the scheme of everything it happens more often than not, but in the sense that it’s quite difficult to create rings which aren’t isomorphic to their opposite ring. Probably the easiest example is if one takes \text{End}\left(\mathbb{Z}\times\mathbb{Q}\right) (endomorphism ring as an abelian group) but it’s not in my interest to prove this.

\text{ }

We define an antihomomomorphism between two rings R and S to be a map R\to S such that the induced map R\to S^\text{op} is a homomorphism.  In other words, it’s a map f:R\to S which is a group homomorphism and for which f(ab)=f(b)f(a).

\text{ }

\text{ }

References:

1. Dummit, David Steven., and Richard M. Foote. Abstract Algebra. Hoboken, NJ: Wiley, 2004. Print.

2. Bhattacharya, P. B., S. K. Jain, and S. R. Nagpaul. Basic Abstract Algebra. Cambridge [Cambridgeshire: Cambridge UP, 1986. Print.

Advertisements

June 18, 2011 - Posted by | Algebra, Ring Theory | , , ,

1 Comment »

  1. […] and “compose in the opposite way”. In this sense it has a very similar feel to the opposite ring. This category is interesting because it allows us to talk about contravariant functors as […]

    Pingback by The Opposite Category « Abstract Nonsense | December 30, 2011 | Reply


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

%d bloggers like this: