Endomorphism rings of torsionless modules
Author:
Arun Vinayak Jategaonkar
Journal:
Trans. Amer. Math. Soc. 161 (1971), 457466
MSC:
Primary 16.40
MathSciNet review:
0284464
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Abstract: Let A be a right order in a semisimple ring be a finitedimensional torsionless right Amodule and be the injective hull of M. J. M. Zelmanowitz has shown that is a semisimple ring and is a right order in Q. Further, if A is a twosided order in then S is a twosided order in Q. We give a conceptual proof of this result. Moreover, we show that if A is a bounded order then so is S. The underlying idea of our proofs is very simple. Rather than attacking directly, we prove the results for . If is the projection on M along then, of course, and it is easy to transfer the required information from B to S. The reason why it is any easier to look at B rather than S is that is a generator in  and a Morita type transfer of properties from A to B is available. We construct an Artinian ring resp. Noetherian prime ring containing a right ideal whose endomorphism ring fails to be Artinian resp. Noetherian from either side.
 [1]
S.
A. Amitsur, Rings of quotients and Morita contexts, J. Algebra
17 (1971), 273–298. MR 0414604
(54 #2704)
 [2]
Hyman
Bass, Algebraic 𝐾theory, W. A. Benjamin, Inc., New
YorkAmsterdam, 1968. MR 0249491
(40 #2736)
 [3]
Carl
Faith, A correspondence theorem for
projective modules and the structure of simple noetherian rings,
Bull. Amer. Math. Soc. 77 (1971), 338–342. MR 0280534
(43 #6254), http://dx.doi.org/10.1090/S000299041971126848
 [4]
R.
Hart, Endomorphisms of modules over semiprime rings, J.
Algebra 4 (1966), 46–51. MR 0194466
(33 #2676)
 [5]
Nathan
Jacobson, The Theory of Rings, American Mathematical Society
Mathematical Surveys, vol. II, American Mathematical Society, New York,
1943. MR
0008601 (5,31f)
 [6]
Arun
Vinayak Jategaonkar, A counterexample in ring theory and
homological algebra, J. Algebra 12 (1969),
418–440. MR 0240131
(39 #1485)
 [7]
Lawrence
Levy, Torsionfree and divisible modules over
nonintegraldomains, Canad. J. Math. 15 (1963),
132–151. MR 0142586
(26 #155)
 [8]
Lance
W. Small, Orders in Artinian rings. II, J. Algebra
9 (1968), 266–273. MR 0230755
(37 #6315)
 [9]
Julius
Martin Zelmanowitz, Endomorphism rings of torsionless modules,
J. Algebra 5 (1967), 325–341. MR 0202766
(34 #2626)
 [1]
 S. A. Amitsur, Rings of quotients and Morita context, J. Algebra (to appear). MR 0414604 (54:2704)
 [2]
 H. Bass, Algebraic Ktheory, Benjamin, New York, 1968. MR 40 #2736. MR 0249491 (40:2736)
 [3]
 C. Faith, The correspondence theorem for projective modules and the structure of simple Noetherian rings, Bull. Amer. Math. Soc. 77 (1971), 338342. MR 0280534 (43:6254)
 [4]
 R. Hart, Endomorphisms of modules over semiprime rings, J. Algebra 4 (1966), 4651. MR 33 #2676. MR 0194466 (33:2676)
 [5]
 N. Jacobson, The theory of rings, Math. Surveys, no. II, Amer. Math. Soc., Providence, R. I., 1943. MR 5, 31. MR 0008601 (5:31f)
 [6]
 A. V. Jategaonkar, A counterexample in ring theory and homological algebra, J. Algebra 12 (1969), 418440. MR 39 #1485. MR 0240131 (39:1485)
 [7]
 L. Levy, Torsionfree and divisible modules over nonintegraldomains, Canad. J. Math. 15 (1963), 132151. MR 0142586 (26:155)
 [8]
 L. W. Small, Orders in Artinian rings. II, J. Algebra 9 (1968), 266273. MR 37 #6315. MR 0230755 (37:6315)
 [9]
 J. M. Zelmanowitz, Endomorphism rings of torsionless modules, J. Algebra 5 (1967), 325341. MR 34 #2626. MR 0202766 (34:2626)
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Additional Information
DOI:
http://dx.doi.org/10.1090/S00029947197102844649
PII:
S 00029947(1971)02844649
Keywords:
Finitedimensional modules,
torsionless modules,
nonsingular modules,
generators,
endomorphism rings,
orders in semisimple rings,
bounded orders,
Morita theorems,
prime Noetherian rings,
Artinian ring
Article copyright:
© Copyright 1971
American Mathematical Society
