Generalizations of 𝑄𝐹-3 algebras

Colby, R. R.; Rutter, E. A.

doi:10.1090/S0002-9947-1971-0269686-5

Generalizations of $\textrm {QF}-3$ algebras
HTML articles powered by AMS MathViewer

by R. R. Colby and E. A. Rutter PDF

Trans. Amer. Math. Soc. 153 (1971), 371-386 Request permission

Abstract:

This paper consists of three parts. The first is devoted to investigating the equivalence and left-right symmetry of several conditions known to characterize finite dimensional algebras which have a unique minimal faithful representation— QF-$3$ algebras—in the class of left perfect rings. It is shown that the following conditions are equivalent and imply their right-hand analog: $R$ contains a faithful $\Sigma$-injective left ideal, $R$ contains a faithful $II$-projective injective left ideal; the injective hulls of projective left $R$-modules are projective, and the projective covers of injective left $R$-modules are injective. Moreover, these rings are shown to be semi-primary and to include all left perfect rings with faithful injective left and right ideals. The second section is concerned with the endomorphism ring of a projective module over a hereditary or semihereditary ring. More specifically we consider the question of when such an endomorphism ring is hereditary or semihereditary. In the third section we establish the equivalence of a number of conditions similar to those considered in the first section for the class of hereditary rings and obtain a structure theorem for this class of hereditary rings. The rings considered are shown to be isomorphic to finite direct sums of complete blocked triangular matrix rings each over a division ring.

References

Felix Albrecht, On projective modules over semi-hereditary rings, Proc. Amer. Math. Soc. 12 (1961), 638–639. MR 126470, DOI 10.1090/S0002-9939-1961-0126470-X
Maurice Auslander, On the dimension of modules and algebras. III. Global dimension, Nagoya Math. J. 9 (1955), 67–77. MR 74406, DOI 10.1017/S0027763000023291
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
Henri Cartan and Samuel Eilenberg, Homological algebra, Princeton University Press, Princeton, N. J., 1956. MR 0077480
P. M. Cohn, Quadratic extensions of skew fields, Proc. London Math. Soc. (3) 11 (1961), 531–556. MR 136633, DOI 10.1112/plms/s3-11.1.531
R. R. Colby and Edgar A. Rutter Jr., Semi-primary $\textrm {QF}-3$ rings, Nagoya Math. J. 32 (1968), 253–258. MR 230762, DOI 10.1017/S0027763000026672
Robert R. Colby and Edgar A. Rutter Jr., Semi-perfect $\textrm {QF}$-$3$ and $\textrm {PP}$-rings, Osaka Math. J. 5 (1968), 99–102. MR 233847
R. R. Colby and E. A. Rutter Jr., $\textrm {QF}-3$ rings with zero singular ideal, Pacific J. Math. 28 (1969), 303–308. MR 244318, DOI 10.2140/pjm.1969.28.303
B. Eckmann and A. Schopf, Über injektive Moduln, Arch. Math. (Basel) 4 (1953), 75–78 (German). MR 55978, DOI 10.1007/BF01899665
Samuel Eilenberg, Homological dimension and syzygies, Ann. of Math. (2) 64 (1956), 328–336. MR 82489, DOI 10.2307/1969977
Carl Faith, Lectures on injective modules and quotient rings, Lecture Notes in Mathematics, No. 49, Springer-Verlag, Berlin-New York, 1967. MR 0227206, DOI 10.1007/BFb0074319
Carl Faith, Rings with ascending condition on annihilators, Nagoya Math. J. 27 (1966), 179–191. MR 193107, DOI 10.1017/S0027763000011983
Carl Faith and Elbert A. Walker, Direct-sum representations of injective modules, J. Algebra 5 (1967), 203–221. MR 207760, DOI 10.1016/0021-8693(67)90035-X
Kent R. Fuller, The structure of $\textrm {QF}-3$ rings, Trans. Amer. Math. Soc. 134 (1968), 343–354. MR 227225, DOI 10.1090/S0002-9947-1968-0227225-9
Kent R. Fuller, On indecomposable injectives over artinian rings, Pacific J. Math. 29 (1969), 115–135. MR 246917, DOI 10.2140/pjm.1969.29.115
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
A. W. Goldie, Torsion-free modules and rings, J. Algebra 1 (1964), 268–287. MR 164991, DOI 10.1016/0021-8693(64)90023-7
Manabu Harada, On semi-primary $PP$-rings, Osaka Math. J. 2 (1965), 153–161. MR 204457

and semi-primary

-rings

3

34

Nathan Jacobson, Structure of rings, Revised edition, American Mathematical Society Colloquium Publications, Vol. 37, American Mathematical Society, Providence, R.I., 1964. MR 0222106
J. P. Jans, Projective injective modules, Pacific J. Math. 9 (1959), 1103–1108. MR 112904, DOI 10.2140/pjm.1959.9.1103
James P. Jans, Rings and homology, Holt, Rinehart and Winston, New York, 1964. MR 0163944
Irving Kaplansky, Projective modules, Ann. of Math (2) 68 (1958), 372–377. MR 0100017, DOI 10.2307/1970252
Lawrence Levy, Torsion-free and divisible modules over non-integral-domains, Canadian J. Math. 15 (1963), 132–151. MR 142586, DOI 10.4153/CJM-1963-016-1
H. Y. Mochizuki, On the double commutator algebra of $QF-3$ algebras, Nagoya Math. J. 25 (1965), 221–230. MR 177007, DOI 10.1017/S0027763000011557
Kiiti Morita, Duality for modules and its applications to the theory of rings with minimum condition, Sci. Rep. Tokyo Kyoiku Daigaku Sect. A 6 (1958), 83–142. MR 96700

Duality in

rings

108

Bruno J. Müller, Dominant dimension of semi-primary rings, J. Reine Angew. Math. 232 (1968), 173–179. MR 233854, DOI 10.1515/crll.1968.232.173
B. L. Osofsky, Rings all of whose finitely generated modules are injective, Pacific J. Math. 14 (1964), 645–650. MR 161886, DOI 10.2140/pjm.1964.14.645
B. L. Osofsky, A generalization of quasi-Frobenius rings, J. Algebra 4 (1966), 373–387. MR 204463, DOI 10.1016/0021-8693(66)90028-7
B. L. Osofsky, Noninjective cyclic modules, Proc. Amer. Math. Soc. 19 (1968), 1383–1384. MR 231857, DOI 10.1090/S0002-9939-1968-0231857-7
Francis L. Sandomierski, Semisimple maximal quotient rings, Trans. Amer. Math. Soc. 128 (1967), 112–120. MR 214624, DOI 10.1090/S0002-9947-1967-0214624-3
Francis L. Sandomierski, Nonsingular rings, Proc. Amer. Math. Soc. 19 (1968), 225–230. MR 219568, DOI 10.1090/S0002-9939-1968-0219568-5
Lance W. Small, Semihereditary rings, Bull. Amer. Math. Soc. 73 (1967), 656–658. MR 212051, DOI 10.1090/S0002-9904-1967-11812-3
Hiroyuki Tachikawa, On left $\textrm {QF}-3$ rings, Pacific J. Math. 32 (1970), 255–268. MR 257148, DOI 10.2140/pjm.1970.32.255
R. M. Thrall, Some generalization of quasi-Frobenius algebras, Trans. Amer. Math. Soc. 64 (1948), 173–183. MR 26048, DOI 10.1090/S0002-9947-1948-0026048-0
Kenneth G. Wolfson, Baer rings of endomorphisms, Math. Ann. 143 (1961), 19–28. MR 122842, DOI 10.1007/BF01351889