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Let be a commutative ring and let be a nonzero proper ideal of . In this paper, we study the properties of a family of rings , with , as quotients of the Rees algebra , when is a semidualizing ideal of Noetherian ring , and in the case that is a flat ideal of . In particular, for a Noetherian ring , it is shown that if is a finitely generated -module, then is totally -reflexive as an -module if and only if is totally reflexive as an -module, provided that is a semidualizing ideal and is reducible in . In addition, it is proved that if is a nonzero flat ideal of and is reducible in , then fdR(M) = fdR(I)a,b(M ⊗RR(I)a,b) = fdR(M ⊗RR(I)a,b), for any -module .