This definition according to the abstract of Madsen, Thomas & Wall 1983. Other authors take the smallest $e(G)$ such that $e(G) \cdot \chi$ is in this ideal for all rational representations $\chi$ (e.g. Yamauchi 70), in which case $e(G)$ is the exponent of the additive group underlying the quotient by the ideal, whence the terminology.

Kenichi Yamauchi, On the 2-part of Artin Exponent of Finite Groups, Science Reports of the Tokyo Kyoiku Daigaku, Section A Vol. 10, No. 263/274 (1970), 234-240 (jstor:43698746)

K. K. Nwabueze, Some definition of the Artin exponent of finite groups (arXiv:math/9611212)

S. Jafari and H. Sharifi, On the Artin exponent of some rational groups, Communications in Algebra, 46:4, 1519-152 (doi:10.1080/00927872.2017.1347665)

Textbook account:

Charles Curtis, Irving Reiner, Methods of representation theory – With applications to finite groups and orders – Vol. I, Wiley 1981 (pdf)