This behaviour comes from the TeX-insert-dollar function which is described as
(defun TeX-insert-dollar (&optional arg)
"Insert dollar sign.
If current math mode was not entered with a dollar, refuse to
insert one. Show matching dollar sign if this dollar sign ends
the TeX math mode and `blink-matching-paren' is non-nil.
When outside math mode, the behavior is controlled by the variable
`TeX-electric-math'.
With raw \[universal-argument] prefix, insert exactly one dollar
sign. With optional ARG, insert that many dollar signs."
Looking at the implementation there does not seem to be a possibility to only disable this specific behaviour of refusing to insert $. You basically have two options:
Follow the documentation shown above and use the universal argument, i.e. press C-u $ to insert a dollar sign.
Remove the $ key from the mode-dependent mapping, i.e.
(define-key TeX-mode-map "$" nil)
This will map $ back to self-insert-command (or whatever is in your global map) but this way you will also lose the other functionality brought by TeX-insert-dollar such as TeX-electric-math.
$starts or ends math mode and\$is needed to typeset a$? – David Carlisle Jan 14 '21 at 23:19cases*environment wants text in the second column; thus it's a real AUCTeX error not to allow for$in that position. Apparently, AUCTeX hasn't got support for themathtoolsextensions toamsmath. – egreg Jan 15 '21 at 06:53cases*can be turned into no-math but this applies only to half of the env. One can always useC-q $to insert a character (here$), it seems to me the easiest way. – Arash Esbati Jan 16 '21 at 11:05$there, not\$like it was suggested. – egreg Jan 16 '21 at 11:45