A remark on a theorem of Y. Kurata

In [K] Y.Kurata proved that the Goldie torsion theory splits centrally for dual rings. Here we extend his result to semilocal rings with left essential socle such that Soc (RR) ⊆ Soc (RR). An example will demonstrate that our observation extends Kurata’s result. All rings are associative ring with unit and all R-modules are unital. The singular submodule of a left R-module M is denoted by Z(RM). We abbreviate S := Soc (RR) and J := Jac (R) for the left socle resp. the Jacobson radical. We denote the left Goldie torsion theory, that is the hereditary torsion theory generated by all singular left R-modules, by τG and the torsion submodule of a module M by τG(M). τG is said to be centrally splitting if τG(R) is a ring direct summand of R. A classical result of Allin and Dickson [AD] states that τG is centrally splitting for a ring R if and only if R is a direct product of a semisimple ring and a ring with essential left singular ideal. Lemma 1. Let R be a ring with essential left socle S. Then (1) S = S ⊕ (S ∩ Z(RR)), where S is projective and R/S is τG-torsion. (2) J is τG-torsion if and only if S J = 0. Proof. The socle can be decomposed as S = S0 ⊕ S1 where S1 := S ∩ Z(RR) and S0 is a projective left R-module. S ⊆ S0, because for x, y ∈ S with x = x0+x1 and y = y0+y1 where x0, y0 ∈ S0 and x1, y1 ∈ S1. The product xy = xy0 ∈ S0 as xy1 ∈ SZ(RR) = 0. Thus S ⊆ S0 holds and there exists a left module S such that S0 = S ⊕ S. We have SS ⊆ S∩ S = 0. If RS is essential in RR, then S becomes singular (as it is annihilated by S) and must be zero as it is also projective. Thus S0 = S . Also R/S becomes τG-torsion as S1 ' S/S and R/S are 1991 Mathematics Subject Classification. 16S90.