Etymology
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redemption (n.)

mid-14c., redemcioun, "deliverance from sin," from Old French redemcion (12c.) and directly from Latin redemptionem (nominative redemptio) "a buying back or off, a releasing, a ransoming" (also "bribery"), noun of action from past-participle stem of redimere "to redeem, buy back," from red- "back" (see re-) + emere "to take, buy, gain, procure" (from PIE root *em- "to take, distribute").

The -d- is from the Old Latin habit of using red- as the form of re- before vowels, as also preserved in redact, redolent, redundant. The general sense of "release, repurchase, deliverance" is from late 15c. Commercial sense is from late 15c. Year of Redemption as "Anno Domini" is from 1510s. In the Mercian hymns, Latin redemptionem is glossed by Old English alesnisse.

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Redemptorist (n.)

member of the Congregation of the Most Holy Redeemer (founded Naples, 1732, by St. Alphonsus Liguori), 1835 in English; see redemption. Noted for their missionary work among the poor. Fem. form is Redemptoristine.

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redemptioner (n.)

"one who is or may be redeemed or set at liberty," specifically, in U.S. history, "indentured servant," 1775, from redemption + -er (1).

REDEMPTIONER. One who redeems himself or purchases his release from debt or obligation to the master of a ship by his services; or one whose services are sold to pay the expenses of his passage to America. [Webster, 1830]
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redemptive (adj.)

"redeeming, serving to redeem," 1640s, from redempt (mid-15c.), adjective from Latin redemptus, past participle of redimere "to redeem, buy back" (see redemption) + -ive. Related: Redemptively. Other adjective forms were redemptory (1590s), redemptoric (1889); redemptorial (1823).

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Irredentist (n.)
1882, member of Italian political party formed 1878 which demanded the annexation of neighboring regions where a part of the population was Italian-speaking (Trieste, South Tyrol, Nice, Corsica, etc.); from Italian Irredentista, from irredenta (Italia) "unredeemed (Italy)," fem. of irredento, from in- "not, opposite of" (see in- (1)) + redento, from Latin redemptus, past participle of redimere (see redemption). Related: Irredentism.
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ransom (n.)

13c., raunsoun, "sum paid for the release of a prisoner or captured man," also "redemption from damnation," from Old French ranson (Modern French rançon), earlier raenson "ransom, redemption," from Latin redemptionem (nominative redemptio) "a redeeming," from redimere "to redeem, buy back," from red- "back" (see re-) + emere "to take, buy, gain, procure" (from PIE root *em- "to take, distribute"). A doublet of redemption. A faded word somewhat revived by Scott early 19c. Spelling with -m appears by late 14c., but the reason for it is unclear (compare seldom, random).

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*em- 

Proto-Indo-European root meaning "to take, distribute." 

It forms all or part of: assume; consume; emption; example; exemplar; exemplary; exemplify; exempt; exemption; impromptu; peremptory; pre-emption; premium; presume; presumption; prompt; pronto; ransom; redeem; redemption; resume; sample; sejm; subsume; sumptuary; sumptuous; vintage.

It is the hypothetical source of/evidence for its existence is provided by: Sanskrit yamati "holds, subdues;" Latin emere "buy," originally "take," sumere "to take, obtain, buy;" Old Church Slavonic imo "to take;" Lithuanian imu, imti "to take."

For the sense shift from "take" to "buy" in the Latin verbs, compare Old English sellan "to give," source of Modern English sell "to give in exchange for money;" Hebrew laqah "he bought," originally "he took;" and colloquial English I'll take it for "I'll buy it." 

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re- 

word-forming element meaning "back, back from, back to the original place;" also "again, anew, once more," also conveying the notion of "undoing" or "backward," etc. (see sense evolution below), c. 1200, from Old French re- and directly from Latin re- an inseparable prefix meaning "again; back; anew, against."

Watkins (2000) describes this as a "Latin combining form conceivably from Indo-European *wret-, metathetical variant of *wert- "to turn." De Vaan says the "only acceptable etymology" for it is a 2004 explanation which reconstructs a root in PIE *ure "back."

In earliest Latin the prefix became red- before vowels and h-, a form preserved in redact, redeem, redolent, redundant, redintegrate, and, in disguise, render (v.). In some English words from French and Italian re- appears as ra- and the  following consonant is often doubled (see rally (v.1)).

The many meanings in the notion of "back" give re- its broad sense-range: "a turning back; opposition; restoration to a former state; "transition to an opposite state." From the extended senses in "again," re- becomes "repetition of an action," and in this sense it is extremely common as a formative element in English, applicable to any verb. OED writes that it is "impossible to attempt a complete record of all the forms resulting from its use," and adds that "The number of these is practically infinite ...."   

Often merely intensive, and in many of the older borrowings from French and Latin the precise sense of re- is forgotten, lost in secondary senses, or weakened beyond recognition, so that it has no apparent semantic content (receive, recommend, recover, reduce, recreate, refer, religion, remain, request, require). There seem to have been more such words in Middle English than after, e.g. recomfort (v.) "to comfort, console; encourage;" recourse (n.) "a process, way, course." Recover in Middle English also could mean "obtain, win" (happiness, a kingdom, etc.) with no notion of getting something back, also "gain the upper hand, overcome; arrive at;" also consider the legal sense of recovery as "obtain (property) by judgment or legal proceedings." 

And, due to sound changes and accent shifts, re- sometimes entirely loses its identity as a prefix (rebel, relic, remnant, restive, rest (n.2) "remainder," rally (v.1) "bring together"). In a few words it is reduced to r-, as in ransom (a doublet of redemption), rampart, etc.

It was used from Middle English in forming words from Germanic as well as Latin elements (rebuild, refill, reset, rewrite), and was used so even in Old French (regret, regard, reward, etc.).

Prefixed to a word beginning with e, re- is separated by a hyphen, as re-establish, re-estate, re-edify, etc. ; or else the second e has a dieresis over it: as, reëstablish, reëmbark, etc. The hyphen is also sometimes used to bring out emphatically the sense of repetition or iteration : as, sung and re-sung. The dieresis is not used over other vowels than e when re is prefixed : thus, reinforce, reunite, reabolish. [Century Dictionary, 1895]
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universalism (n.)
1805 in theology, "the doctrine of universal salvation," from universal (adj.) + -ism. Universalist "one who, professing the Christian faith, believes in the eventual redemption of all humanity" is attested from 1620s.
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redeemable (adj.)

"capable of being redeemed" in any sense; "admitting of redemption; capable of being paid off," 1610s, from redeem + -able. As a noun (redeemables) "redeemable property, etc.," by 1720.

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