Set of all $a\in\mathbb Z$ that are coprime to $b\in\mathbb Z$

Question

$\newcommand\Z{\mathbb{Z}}$I'm looking for a standard or reasonable notation for

$$ \{a\in\Z : a\perp b\} = \{a\in\Z : \gcd(a,b)=1\} $$

My problem is that:

• It is not $\Z/b\Z^*$, since for me, $1\neq b+1$.
• It cannot be $\Z_b$, since that are the $b$-adic integers.

So I'm unable to find something fitting. It is not crucial (I can write it out every time), but it would be helpful. Any ideas?

Answer 1

I suppose you could write this as $$ \bigcup \left(\mathbb Z\big/b\mathbb Z\right)^*, $$ where the union just throws the invertible cosets in $\mathbb Z\big/b\mathbb Z$ together.

Anyway, this isn't standard in any way, nor is it more comprehensible than defining some notation like $$ \mathcal C_b = \{\,a\in\mathbb Z \mid \gcd(a,b)=1\,\} $$ on your own. Where $\mathcal C$ is supposed to suggest coprime.

You could also just write "Let $a\in\mathbb Z$ with $(a,b)=1$", since $(a,b)$ is often used to denote $\gcd(a,b)$ in this context.

Answer 2

I don't think there's a standard notation for the set of representatives of the invertible elements under a quotient map of rings.

The set is the union of all the arithmetic progressions ${\cal P}_b(a)=a+{\Bbb Z}b$ as $a$ varies in a set of representatives of $(\Bbb Z/b\Bbb Z)^\times$. If you want a symbol for that, an option could be $$ \cal P_b^\times=\bigcup_{a\in(\Bbb Z/b\Bbb Z)^\times}{\cal P}_b(a). $$