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Nicholson defined a ring to be clean if every element is the sum of a unit and an idempotent. A module is clean if its endomorphism algebra is clean. We show that torsion-complete Abelian p-groups are clean and characterize the clean groups among the class of totally projective p-groups. An example is given of a clean p-group which is neither totally projective nor torsion- complete
Goldsmith, Brendan and P. Vamos: A note on clean abelian groups. Rendiconti del Seminario Matematico della Universita di Padova, 117, 2007, pp.181-191.