expedition to ganges was for territorial reasons part of dig vijaya not just a punitive raid....
we have titles janagai cholan, vadamadurai and kadram kondan for rajendra cholas, successors, rajadhi raja, rajendra chola - 2 , veera rajendra chola.........
greatest of cholas might be vira rajendra chola
cholas did help bhojaa( rajasthan king) greatest of north west indian kingdom) to recover from their loss to chalukyas,he helped him to capture haryana, panchala aka punjab in india& pakistan, adhinagar 9aka srinagar) kingdom , they even led expeditions to peshawar .
invaders tried to enter india from 650 ad but broke out only in 1200 ad to capture delhi.their occupation were mostly thwarted by south indian kingdoms.chalukyas stopped arab invasion in sind , balochistan , pulekesi of sivagamiyin sabatham(tamil novel) gave protection to parsis aka persians, chinese embassy in those days were in contact with pallavas to prevent any arabian advance.
tamil might be aware of this story ,vikramaditya chalukya who took kanchipuram to take revenge for badami loss to narashima pallavan 100 years earlier defeated arab caliphate around 750 ad, this was the time when caiphate was said to be greatest power in the world , they had spain their possession then .
his north indian counterparts gujara prathiharas went further they sent expeditions to afganisthan defeated arab representative and setup a hindu kingdom in its place . invaders broke through by 1200, but south india was practically independent throught with foreighn rule only for about 50 years.Arab caliphate dropped plans of indian invasion