The suffix -ment is attached to verbs to make nouns. The suffix -ment traces back to the Latin noun suffix -mentum.
For exampleLatin: frangere (verb) + -mentum (suffix) = fragmentum
The English noun fragment comes from the Latin noun fragmentum "a fragment, remnant," a combination of the verb frangere "to break" and the suffix -mentum. (Cf. the noun regiment comes from the Latin noun regimentum "rule, direction," from the verb regimen.)
Latin to English: fragmentum (a prodictive morpheme) became fragment (a fused morpheme).
Frag wasn't a verb in English until 1970:
Frag as a verb is first attested 1970 in U.S. military slang, from fragmentation grenade (1918).The oddity about the fused morphemes fragment and regiment
Frag is a new verb (from the noun fragmentation), making fragment (a previously fused morpheme) appear productive:
Modern English: frag (verb) + -ment (suffix)
§ 29. -ment. 8. Word Formation. The American Heritage Book of English Usage. 1996
Online Etymology Dictionary