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I'm working through a proof (Bronstein, Symbolic Integration I, p.139) which uses generator notation. I am familiar with the following:

$\langle t \rangle = \{ t^k : k \in \mathbb{Z} \} $, where $t\in A$.

However, the material with which I am working uses the following

$K\langle t \rangle$,

where $K$ is a field. Is this some kind of generator notation?

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Which book are you using? As notation often depends on the context, you'll get better answers if you tell us, what you are reading. –  martini Jul 27 '12 at 8:31

2 Answers 2

In Basic Algebra 1 by Jacobson, he dosen't use the notation $K<t\ >$, but he does give what you are talking about as an example. It is a monoid, and not, nessacarilly, a group. A monoid is a structure that satisfies all of the group axioms, except for possibly inverses. So, consider it the notation for a monoid generated by t. Of course, the cyclic group is also a monoid, however.

For a specific example, look at the integers under addition. The monoid generated b 1 would be the natural numbers, with zero. Moreover, this is not a group. However, if you were working in $\mathbb{Z}_n$, under addition. In this case, all inverses can be written as positive 'powers.'

For an example, modulo 6, the inverse of two is 4, which is 2+2 (to the power of two, I know that you know that 2+2=4 [lol])

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up vote 0 down vote accepted

If $(k,D)$ is a differential field then we have:

$\quad\quad k\langle t\rangle = \left\{\begin{array}{ll} k[t] & : Dt\in k \\ k[t,t^{-1}] & : Dt/t\in k \end{array}\right..$

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