pattern - What is a Decomposed Word™?

_{This is in the spirit of the What is a Word/Phrase™ series started by JLee with a special brand of Phrase™ and Word™ puzzles.}

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If a word has a certain property, I call it a Decomposed Word™.

You can use the examples below to find the property:

$$\begin{array}{|c|c|} \hline \bbox[yellow]{\textbf{Decomposed Words™}}&\bbox[yellow]{\textbf{Not Decomposed Words™}}\\ \hline \text{JOINING}&\text{ADHERING}\\ \hline \text{AGGRAVATE}&\text{EXACERBATE}\\ \hline \text{ANNEXING}&\text{APPENDING}\\ \hline \text{WATCHLIST}&\text{REGISTER}\\ \hline \text{VANQUISHING}&\text{DEFEATING}\\ \hline \text{PLAINTEXT}&\text{CLEARTEXT}\\ \hline \text{DESEXUALIZE}&\text{EMASCULATE}\\ \hline \text{MOCCASIN}&\text{SLIPPER}\\ \hline \text{LITIGATION}&\text{LAWSUIT}\\ \hline \text{MIX}&\text{STIR}\\ \hline \text{CODIFY}&\text{TABULATE}\\ \hline \text{METHAMPHETAMINE}&\text{OPIOID}\\ \hline \text{CONQUERING}&\text{SUBJUGATING}\\ \hline \end{array} $$

For those without MathJax, or if you want to pop this into a spreadsheet, here is a CSV version:

Decomposed™,Not Decomposed™
JOINING,ADHERING
AGGRAVATE,EXACERBATE
ANNEXING,APPENDING
WATCHLIST,REGISTER
VANQUISHING,DEFEATING
PLAINTEXT,CLEARTEXT

DESEXUALIZE,EMASCULATE
MOCCASIN,SLIPPER
LITIGATION,LAWSUIT
MIX,STIR
CODIFY,TABULATE
METHAMPHETAMINE,OPIOID
CONQUERING,SUBJUGATING

Answer

I think Decomposed Words are

those for which, if you extract the letters that are used for Roman numerals, make a prime number

and Not Decomposed Words are accordingly

those which yield either a composite number or 1.

The given examples don't let us tell

how to classify words that contain no such letters.

Per GentlePurpleRain's suggestion in comments: the name is because

when you decompose a number as far as possible into factors, the components you end up with are the prime numbers (what about 1? I hear you cry; well, when you decompose 1 into factors you get the empty set of factors, all of whose elements are in fact prime).

And here are the actual values:

2 II JOINING | 501 DI ADHERING 5 V AGGRAVATE | 90 XC EXACERBATE 11 XI ANNEXING | 501 DI APPENDING 151 CLI WATCHLIST | 1 I REGISTER 7 VII VANQUISHING | 501 DI DEFEATING 59 LIX PLAINTEXT | 160 CLX CLEARTEXT 541 DXLI DESEXUALIZE | 1150 MCL EMASCULATE 1201 MCCI MOCCASIN | 51 LI SLIPPER 53 LIII LITIGATION | 51 LI LAWSUIT 1009 MIX MIX | 1 I STIR 401 CDI CODIFY | 50 L TABULATE 3001 MMMI METHAMPHETAMINE | 498 IID OPIOID 101 CI CONQUERING | 1 I SUBJUGATING

You may notice that the second-last one of those on the right

is nonstandard because you can't really put two smaller "decrementing" Is before a single D. DII is also composite (of course; it's even and doesn't equal 2) but if we just add up Roman numerals without regard to position then MIX no longer works because MXI = 1011 which, being a multiple of 3, is not prime.

But the questioner indicates in comments that

his intention was that anything not containing a valid Roman numeral should be a Not Decomposed Word, so OPIOID is fine for that reason.