We shall consider intelligent homological dimensions of 0-direct union of finite Rees matrix semigroups. First we give some characterizations of Rees matrix semigroup algebras. Then if K is a ring and S has a Rees matrix ideal M, we give the bounds of r.gl.dim K[S] in terms of r.gl.dim K [Ti] and r.gl.dim K[S/M]. The bounds for r.gl.dim K[S] are given, supposing that the finite monoid S is a 0-direct union of some finite Rees matrix semigroups which are in forms of Si = M (Ti ; Ii ; Λi ; Pi). In addition, it is determined that K[S] has finite homological dimension if and only if each Ti has finite homological dimension.
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