What is coarb?

Coarb

A coarb, from the Old Irish comarbae (Modern Irish comharba), meaning "heir" or "successor", was a distinctive office of the later medieval church among the Gaels of Ireland and Scotland. In this period coarb appears interchangeable with "erenach", denoting the episcopally nominated lay guardian of a parish church and headman of the family in hereditary occupation of church lands. The coarb, however, often had charge of a church which had held comparatively high rank in pre‐Norman Ireland, or one still possessed of relatively extensive termon lands.

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Some articles on coarb:

The Coarb of Columba
... In medieval Ireland and Scotland, the coarb of St Columba (Medieval Gaelic comarba Coluim Chille) identified the abbots who succeeded Columba ... following the 9th century Viking raids on Iona, their abbot continued to hold the title of coarb to reflect his direct inheritance many of the early abbots were members of Columba's family ...
Fenagh, County Leitrim - History
... there was no roof on the church of Fenagh, and the coarb was away that night ... tents which were inside the church, without permission of their leaders, and the coarb's foster-child, God's gift, was suffocated ... Now learned men relate that the coarb received this foster-child by finding him on a large stone which stood in that place, and never knew of his having either mother or father and the coarb loved him and gave him ...
Abbots Of Iona - List of Comarbai Coluim Cille and Abbots of Iona, 891–1099
947 Was definitely abbot of Iona, but may not have been coarb of Coluim Cille Robartach Raphoe Died 954 Dub Dúin ua Stepháin unknown Died 959 Dub Scoile mac Cin ... unknown Died 1062 Gilla Críst was called "coarb of Colum Cille in Ireland and Scotland" ...
Clan Mac Lea - Coarb of Saint Moluag
... The Coarb, or successor, of the saint was the hereditary keeper of his pastoral staff ... of Arms ruled that Livingstone was the Coarb of Saint Moluag ... As his successor, the Coarb is viewed as a sovereign baron and is granted a unique Cap of Maintenance of Gules doubled Vair to place in his coat of arms ...
Armagh - Annalistic References
... AI973.3 Dub dá Leithe, coarb of Patrick, came to Mumu and made his visitation and he and the coarb of Ailbe quarrelled regarding the levy, and Mathgamain, king of Mumu, made peace ... AI996.5 Dub dá Leithe, coarb of Ard Macha (or, of Patrick) and coarb of Colum Cille, rested in Christ ... AI1020.3 Mael Muire son of Eochaid, coarb of Patrick, rested in Christ ...