Regular 𝑃.𝐼.-rings

Armendariz, E. P.; Fisher, Joe W.

doi:10.1090/S0002-9939-1973-0313305-3

Regular $P.I.$-rings
HTML articles powered by AMS MathViewer

by E. P. Armendariz and Joe W. Fisher PDF

Proc. Amer. Math. Soc. 39 (1973), 247-251 Request permission

Abstract:

For a ring $R$ which satisfies a polynomial identity we show that the following are equivalent: (1) $R$ is von Neumann regular, (2) each two-sided ideal of $R$ is idempotent, and (3) each simple left (right) $R$-module is injective. We show that a P.I.-ring $R$ is left perfect if and only if all left $R$-modules have maximal submodules and $R$ has no infinite sets of orthogonal idempotents.

References

S. A. Amitsur, Prime rings having polynomial identities with arbitrary coefficients, Proc. London Math. Soc. (3) 17 (1967), 470–486. MR 217118, DOI 10.1112/plms/s3-17.3.470

On reduced

-rings

Gorô Azumaya, Strongly $\pi$-regular rings, J. Fac. Sci. Hokkaido Univ. Ser. I. 13 (1954), 34–39. MR 0067864
Hyman Bass, Finitistic dimension and a homological generalization of semi-primary rings, Trans. Amer. Math. Soc. 95 (1960), 466–488. MR 157984, DOI 10.1090/S0002-9947-1960-0157984-8
V. C. Cateforis and F. L. Sandomierski, On commutative rings over which the singular submodule is a direct summand for every module, Pacific J. Math. 31 (1969), 289–292. MR 249420, DOI 10.2140/pjm.1969.31.289
John H. Cozzens, Homological properties of the ring of differential polynomials, Bull. Amer. Math. Soc. 76 (1970), 75–79. MR 258886, DOI 10.1090/S0002-9904-1970-12370-9
Joe W. Fisher and Robert L. Snider, On the von Neumann regularity of rings with regular prime factor rings, Pacific J. Math. 54 (1974), 135–144. MR 360679, DOI 10.2140/pjm.1974.54.135
A. W. Goldie, Semi-prime rings with maximum condition, Proc. London Math. Soc. (3) 10 (1960), 201–220. MR 111766, DOI 10.1112/plms/s3-10.1.201
Ross M. Hamsher, Commutative, noetherian rings over which every module has a maximal submodule, Proc. Amer. Math. Soc. 17 (1966), 1471–1472. MR 200303, DOI 10.1090/S0002-9939-1966-0200303-X
Irving Kaplansky, Rings with a polynomial identity, Bull. Amer. Math. Soc. 54 (1948), 575–580. MR 25451, DOI 10.1090/S0002-9904-1948-09049-8
Irving Kaplansky, Topological representation of algebras. II, Trans. Amer. Math. Soc. 68 (1950), 62–75. MR 32612, DOI 10.1090/S0002-9947-1950-0032612-4
L. A. Koĭfman, Rings over which each module has a maximal submodule, Mat. Zametki 7 (1970), 359–367 (Russian). MR 262303
J. Levitzki, A theorem on polynomial identities, Proc. Amer. Math. Soc. 1 (1950), 334–341. MR 35758, DOI 10.1090/S0002-9939-1950-0035758-5
G. O. Michler and O. E. Villamayor, On rings whose simple modules are injective, J. Algebra 25 (1973), 185–201. MR 316505, DOI 10.1016/0021-8693(73)90088-4
Edward C. Posner, Prime rings satisfying a polynomial identity, Proc. Amer. Math. Soc. 11 (1960), 180–183. MR 111765, DOI 10.1090/S0002-9939-1960-0111765-5
Guy Renault, Sur les anneaux $A$, tels que tout $A$-module à gauche non nul contient un sous-module maximal, C. R. Acad. Sci. Paris Sér. A-B 264 (1967), A622–A624 (French). MR 214623