Endomorphism rings of projective modules
Author:
Roger Ware
Journal:
Trans. Amer. Math. Soc. 155 (1971), 233256
MSC:
Primary 16.40
MathSciNet review:
0274511
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Abstract: The object of this paper is to study the relationship between certain projective modules and their endomorphism rings. Specifically, the basic problem is to describe the projective modules whose endomorphism rings are (von Neumann) regular, local semiperfect, or left perfect. Call a projective module regular if every cyclic submodule is a direct summand. Thus a ring is a regular module if it is a regular ring. It is shown that many other equivalent ``regularity'' conditions characterize regular modules. (For example, every homomorphic image is flat.) Every projective module over a regular ring is regular and a number of examples of regular modules over nonregular rings are given. A structure theorem is obtained: every regular module is isomorphic to a direct sum of principal left ideals. It is shown that the endomorphism ring of a finitely generated regular module is a regular ring. Conversely, over a commutative ring a projective module having a regular endomorphism ring is a regular module. Examples are produced to show that these results are the best possible in the sense that the hypotheses of finite generation and commutativity are needed. An application of these investigations is that a ring is semisimple with minimum condition if and only if the ring of infinite row matrices over is a regular ring. Next projective modules having local, semiperfect and left perfect endomorphism rings are studied. It is shown that a projective module has a local endomorphism ring if and only if it is a cyclic module with a unique maximal ideal. More generally, a projective module has a semiperfect endomorphism ring if and only if it is a finite direct sum of modules each of which has a local endomorphism ring.
 [1]
Hyman
Bass, Finitistic dimension and a homological
generalization of semiprimary rings, Trans.
Amer. Math. Soc. 95
(1960), 466–488. MR 0157984
(28 #1212), http://dx.doi.org/10.1090/S00029947196001579848
 [2]
, The Morita theorems (Oregon Lectures, 1962).
 [3]
Henri
Cartan and Samuel
Eilenberg, Homological algebra, Princeton University Press,
Princeton, N. J., 1956. MR 0077480
(17,1040e)
 [4]
Stephen
U. Chase, Direct products of modules,
Trans. Amer. Math. Soc. 97 (1960), 457–473. MR 0120260
(22 #11017), http://dx.doi.org/10.1090/S00029947196001202603
 [5]
B.
Eckmann and A.
Schopf, Über injektive Moduln, Arch. Math. (Basel)
4 (1953), 75–78 (German). MR 0055978
(15,5d)
 [6]
Irving
Kaplansky, Projective modules, Ann. of Math (2)
68 (1958), 372–377. MR 0100017
(20 #6453)
 [7]
Irving
Kaplansky, Fields and rings, The University of Chicago Press,
Chicago, Ill.London, 1969. MR 0269449
(42 #4345)
 [8]
Erika
A. Mares, Semiperfect modules, Math. Z. 82
(1963), 347–360. MR 0157985
(28 #1213)
 [9]
F.
L. Sandomierski, On semiperfect and perfect
rings, Proc. Amer. Math. Soc. 21 (1969), 205–207. MR 0237567
(38 #5848), http://dx.doi.org/10.1090/S00029939196902375675
 [10]
Oscar
Zariski and Pierre
Samuel, Commutative algebra, Volume I, The University Series
in Higher Mathematics, D. Van Nostrand Company, Inc., Princeton, New
Jersey, 1958. With the cooperation of I. S. Cohen. MR 0090581
(19,833e)
 [11]
Alex
Rosenberg and Daniel
Zelinsky, Finiteness of the injective hull, Math. Z.
70 (1958/1959), 372–380. MR 0105434
(21 #4176)
 [1]
 H. Bass, Finitistic dimension and a homological generalization of semiprimary rings, Trans. Amer. Math Soc. 95 (1960), 466488. MR 28 #1212. MR 0157984 (28:1212)
 [2]
 , The Morita theorems (Oregon Lectures, 1962).
 [3]
 H. Cartan and S. Eilenberg, Homological algebra, Princeton Univ. Press, Princeton, N. J., 1956. MR 17, 1040. MR 0077480 (17:1040e)
 [4]
 S. U. Chase, Direct products of modules, Trans. Amer. Math. Soc. 97 (1960), 457473. MR 22 #11017. MR 0120260 (22:11017)
 [5]
 B. Eckmann and A. Schopf, Über injektive Moduln, Arch. Math. 4 (1953), 7578. MR 15, 5. MR 0055978 (15:5d)
 [6]
 I. Kaplansky, Projective modules, Ann. of Math. (2) 68 (1958), 372377. MR 20 #6453. MR 0100017 (20:6453)
 [7]
 , Fields and rings, Univ. of Chicago Press, Chicago, Ill., 1969. MR 0269449 (42:4345)
 [8]
 E. A. Mares, Semiperfect modules, Math. Z. 82 (1963), 347360. MR 28 #1213. MR 0157985 (28:1213)
 [9]
 F. L. Sandomierski, On semiperfect and perfect rings, Proc. Amer. Math. Soc. 21 (1969), 205207. MR 38 #5848. MR 0237567 (38:5848)
 [10]
 O. Zariski and P. Samuel, Commutative algebra. Vol. I, University Series in Higher Math., Van Nostrand, Princeton, N. J., 1958. MR 19, 833. MR 0090581 (19:833e)
 [11]
 A. Rosenberg and D. Zelinsky, Finiteness of the injective hull, Math. Z. 70 (1958/59), 372380. MR 21 #4176. MR 0105434 (21:4176)
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Additional Information
DOI:
http://dx.doi.org/10.1090/S00029947197102745112
PII:
S 00029947(1971)02745112
Keywords:
Projective module,
endomorphism ring,
Jacobson radical,
von Neumann regular ring,
local ring,
semiperfect ring,
left perfect ring,
left nilpotent,
flat module,
semiprime ring,
row finite matrices,
injective module
Article copyright:
© Copyright 1971
American Mathematical Society
