Regrettably jailing one giving false name (5)
I believe the answer is:
alias
'false name' is the definition.
(synonyms)
'regrettably jailing one' is the wordplay.
'regrettably' becomes 'alas' ('alas' is said before regrettable information).
'jailing' indicates putting letters inside (inserted letters are jailed).
'one' becomes 'i' (Roman numeral).
'alas' enclosing 'i' is 'ALIAS'.
'giving' acts as a link.
(Other definitions for alias that I've seen before include "Handle that isn't genuine" , "Cabbage salad" , "A sail (anag)" , "False identity" , "Otherwise known as" .)