  For an element in an algebra-valued *-noncommutative probability space,
equivalent conditions for algebra-valued R-diagonality (a notion introduced by
Sniady and Speicher) are proved. Formal power series relations involving the
moments and cumulants of such R-diagonal elements are proved. Decompositions of
algebra-valued R-diagonal elements into products of the form unitary times
self-adjoint are investigated; sufficient conditions, in terms of cumulants,
for *-freeness of the unitary and the self-adjoint part are proved, and a
tracial example is given where *-freeness fails. The particular case of
algebra-valued circular elements is considered; as an application, the polar
decompostion of the quasinilpotent DT-operator is described.
  An erratum is appended. (The last sentence in the above paragraph is thereby
nullified.)
