When every finitely generated flat module is projective

Abstract

We investigate the class of rings over which every finitely generated flat right module is projective.

Normalizing extensions of homogeneous semilocal rings and related rings

“…In [7] Puninski and Rothmaler proved that the FGFP property is closed under Morita equivalence, finite direct sum and subrings. Jondrup [4] proved that FGFP property is interchanged between a ring R and the group ring R[G], also between R and the ring of power series R½½x.…”

“…. of M n ðRÞ the n  n matrices over R such that A iþ1 A i ¼ A i for every i, eventually consists of idempotent generating the same principal right ideal in M n ðRÞ, then we say in this case that the sequence converges [7]. Following Sakhaev [9] an ideal I of a ring R is called weakly commutative if there exists m P 2 such that for all a 1 ; a 2 ; .…”

The generating hypothesis in the derived category of a ring

“…There are a great many rings where finitely generated flat modules are known to be projective [PR04]. The following theorem contains some cases of this, which are somewhat less satisfactory since not all von Neumann regular rings satisfy the hypotheses.…”

When every projective module is a direct sum of finitely generated modules

“…For instance, every commutative semilocal ring is an F -ring, and so is every commutative ring of finite Goldie dimension. Note, the F -rings are precisely the commutative S-rings from [41]. It is well known that the trace of a finitely generated projective module over a commutative ring is generated by an idempotent, see [29, 2.43].…”