The spelling without the medial e, abridgment, is standard in American English; with the e, abridgement, is standard in British English. This habit usually follows when appending suffixes to root words ending in -dge: lodg(e)ment, abridg(e)ment, acknowledg(e)ment, and judg(e)ment all keep the parenthesized e only in British English. But to trouble simplicity, judgment has been preferred in British English writing, though Bryan A. Garner suggests that this may be primarily the habit in legal contexts.
The above rule, annoyingly, does not transpose to all other root words with a final -e. Argument and monument, for example, always lose the -e, while movement, easement, amazement, and abasement (among others) always keep the -e.
While it’s best to learn the differences through repetition, all writers forget how to spell at one point or another. Fortunately, word processors and computers are well-equipped to correct our errors, but do not make a habit of relying on the computer to spell for you; rather than selecting the corrected spelling from the computer’s list of suggestions, make an effort to correct the error yourself and, in so doing, improve your spelling habits.