In general, when appending a suffix to root words ending in -e, the root loses its final -e. Thus argue becomes arguing and argument, create becomes creating and creatable, abate becomes abating and abatable, and nudge becomes nudging and nudged. But verbs ending in -ee, -oe, or -ye tend not to lose their final -e except when appending -ed to form the past tense. Thus we get agreement, dyeing, toeing, and hoed. Exceptions exist, especially for the appended suffix -able, where the medial -e- stays when it indicates pronunciation of a prior vowel: abasement, agreeable, amazement, easement, mineable, sizeable, saleable. Some of these spellings, such as saleable, were once standard only in British English but have become common and preferred in American English.
When appending a suffix to root words ending in -dge, American English usually loses the final -e while British English usually keeps it. Thus abridg(e)ment, acknowledg(e)ment, lodg(e)ment all use the parenthetical e only in British English. To trouble consistency, British English has its exceptions, such as fledgling and judgment, and seems to be slowly adopting the American English spelling in some cases, such as with lodgment, which is now more common than lodgement.
While it’s best to learn the differences through repetition, all writers forget how to spell at one point or another. Fortunately, computers are well-equipped to correct our errors, but do not rely on the computer to spell for you; rather than selecting the corrected spelling from the computer’s list of suggestions, try correcting the error yourself, which will better improve your spelling habits.