Let $A$ be an algebra over a field $F$. It is custom to define for any integer $n$ that $nx=x+...+x$ for any $x\in A$ where the sum has $n$ terms. I wonder if there is a way to make the expression $\frac{1}{n}x$ meaningful? What should, for instance, $\frac{1}{2}[2]X$ mean for $[2]\in \frac{\mathbb{Z}}{3\mathbb{Z}}[X]$, the polynomial algebra with coefficients in $\frac{\mathbb{Z}}{3\mathbb{Z}}$?
"Inverses" in algebras over a field
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linear-algebra
abstract-algebra
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1Simpler question: how can one construe $\frac{1}{2}$ as an element of $F$? – 2017-02-11
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0Are all non-zero elements invertable? – 2017-02-11
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0I'm afraid I don't know how to construct 1/2 as my field $F$ is not the integers and I dont know the meaning of 1/2... – 2017-02-11