On Finitely Generated Flat Modules II.

𝑝.𝑝. rings and finitely generated flat ideals

“…For commutative rings we have that all ideals are flat if the ideals generated by two elements are flat [6 …”

Homological properties of SF rings

“…In particular, any 1-fir (that is, a domain) is left CFP, hence so is also any ring Morita equivalent to it. Note that (c) follows also from [5] as well. Now the same proof used in [13,Theorem 3.1] to show (5) => (1) and (6) => (1) in this reference can be used to prove that, if S satisfies either of the conditions in (d) or (e), then it is left CFP.…”

“…Specifically, (a) follows from the well-known fact that semi-perfect rings are characterised by the statement that every finitely generated module has a projective cover, and from the fact that every projective cover of a flat module is an isomorphism. To see (b), we note that J0ndrup [5] shows that for available at https://www.cambridge.org/core/terms. https://doi.org/10.1017/S0004972700033992 [5] Homological properties of SF rings 331 n-firs, every n-generated flat module is projective.…”

“…To see (b), we note that J0ndrup [5] shows that for available at https://www.cambridge.org/core/terms. https://doi.org/10.1017/S0004972700033992 [5] Homological properties of SF rings 331 n-firs, every n-generated flat module is projective. In particular, any 1-fir (that is, a domain) is left CFP, hence so is also any ring Morita equivalent to it.…”

On the structure of finitely generated splitting rings

“…Since R R is finite dimensional, T = End^(5) is semisimple Artinian. As C is a unital subring of T, by [23,Corollary 3.2] any finitely generated C-flat module is C-projective. It then follows that C is left semihereditary; hence by Small [28,Theorem 3] -flat by [30,Proposition 3.11,p.…”