### Finitely Presented Modules over Right Non-Singular Rings

Ulrich Albrecht
2008 Rendiconti del Seminario Matematico della Universita di Padova
This paper characterizes the right non-singular rings R for which M=Z(M) is projective whenever M is a cyclically (finitely) presented module. Several related results investigate right semi-hereditary rings. b) A a): Let I be the -closure of rR for some r P R. Since R=rR is RDprojective, and R=I (R=rR)=Z(R=rR), we obtain that R=I is projective. Thus, I is generated by an idempotent. By [3, Lemma 3.5], it suffices to show that every -closed right ideal J of R is generated by an idempotent. For
more » ... n idempotent. For this, select 0 T r 0 P J. Since J is -closed in R, it contains the -closure I 0 of r 0 R. By what has been shown so far, I 0 e 0 R for some idempotent e 0 of R. Hence, J e 0 R È [J (1 À e 0 )R]. If J (1 À e 0 )R T 0, select a non-zero r 1 (1 À e 0 )r 1 P J; and observe that J (1 À e 0 )R is -closed in R. Hence, it contains the -closure I 1 of r 1 R in R. By the previous paragraph, I 1 fR for some idempotent f of R. Write f (1 À e 0 )s for some s P R, and set e 1 f (1 À e 0 ). Since e 0 f 0, we have e 1 e 0 e 0 e 1 0 and e 2 1 f 2 À f 2 e 0 À fe 0 f ( fe 0 ) 2 f À fe 0 e 1 . Thus, e 0 and e 1 are non-zero orthogonal idempotents with e 1 R fR. On the other hand, f f (1 À e 0 )s e 1 s yields fR e 1 R. Consequently, R e 0 R È e 1 R È [J (1 À e 0 À e 1 )R]. Continuing inductively, we can construct non-zero orthogonal idempotents e 0 ; . . . ; e n1 P J as long as J (1 À e 0 À . . . À e n )R T 0. Since R does not contain an infinite family of orthogonal idempotent, this process has to stop, say J (1 À e 0 À . . . À e n )R 0. Then, e 0 . . . e m is an idempotent with J (e 0 . . . e m )R. p We now investigate which conditions R has to satisfy to ensure the validity of the structure theorem for pure-projectives in  . We want Finitely Presented Modules over Right Non-Singular Rings