How injective is a semiperfect F.P.F. ring? (English) Zbl 0682.16012
A ring R is called a right FPF ring if every finitely generated faithful right R-module is a generator (in the category of right R-modules). T. G. Faticoni [J. Algebra 107, 297-315 (1987; Zbl 0618.16021)] has shown that a semiperfect right FPF ring has a semiperfect right self- injective classical (left and right) ring of quotients. The current paper extends his investigation into the connection between semiperfect FPF rings and self-injectivity. Let R be a basic semiperfect ring, i.e., R/J(R) is a direct sum of division rings, and let \(R=e_ 1R+...+e_ mR\) be a decomposition into primitive idempotents. The author shows that R is right FPF if and only if (i) every partial homomorphism from \(e_ iR\) to \(e_ jR\) (i\(\neq j)\) extends to a homomorphism from \(e_ iR\) to \(e_ jR\), (ii) every partial homomorphism f on each \(e_ iR\) extends to an endomorphism in \(e_ iR\), provided f is not monic, (iii) given partial homomorphisms f, g on \(e_ iR\) then either \(f=gr\) or \(g=fr\) for some r, (iv) each \(e_ iR\) is uniform, and (v) each nonzero right ideal of R contains a nonzero two-sided ideal.
Reviewer: J.Clark
MSC:
16L60 | Quasi-Frobenius rings |
16L30 | Noncommutative local and semilocal rings, perfect rings |
16D50 | Injective modules, self-injective associative rings |
16W20 | Automorphisms and endomorphisms |
16U99 | Conditions on elements |
Keywords:
semiperfect FPF rings; self-injectivity; basic semiperfect ring; direct sum of division rings; primitive idempotents; endomorphism; partial homomorphismsCitations:
Zbl 0618.16021References:
[1] | DOI: 10.1090/S0002-9939-1976-0417237-4 · doi:10.1090/S0002-9939-1976-0417237-4 |
[2] | DOI: 10.1090/S0002-9939-1977-0429990-5 · doi:10.1090/S0002-9939-1977-0429990-5 |
[3] | Faith Carl, F.P.F. Ring Theory (1984) |
[4] | DOI: 10.1016/0021-8693(87)90092-5 · Zbl 0618.16021 · doi:10.1016/0021-8693(87)90092-5 |
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