Rings whose modules are projective over endomorphism rings
Author:
Robert L. Snider
Journal:
Proc. Amer. Math. Soc. 46 (1974), 164-168
MSC:
Primary 16A50
DOI:
https://doi.org/10.1090/S0002-9939-1974-0347901-5
MathSciNet review:
0347901
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Abstract: Let $R$ be either an Artin ring or a commutative ring. We show that every $R$-module is projective over its endomorphism ring if and only if $R$ is uniserial. We study rings, all of whose modules are projective over their endomorphism rings. We show that for Artin or commutative rings this is equivalent to being uniserial. We remark that we know of no examples other than uniserial rings and conjecture that all such rings are uniserial. Our proof uses the complete ring of quotients as in [4]. Our work is partially motivated by the work of Sally and Vasconcelos on commutative rings whose ideals are projective over their endomorphism rings [7]. We use $J$ throughout to denote the Jacobson radical and $\operatorname {Soc} (R)$ to denote the socle of $R$.
- Vlastimil Dlab and Claus Michael Ringel, Balanced rings, Lectures on rings and modules (Tulane Univ. Ring and Operator Theory Year, 1970–1971, Vol. I), Springer, Berlin, 1972, pp. 73–143. Lecture Notes in Math., Vol. 246. MR 0340344
- Carl Faith, Modules finite over endomorphism ring, Lectures on rings and modules (Tulane Univ. Ring and Operator Theory Year, 1970-1971, Vol. I), Springer, Berlin, 1972, pp. 145–189. Lecture Notes in Math., Vol. 246. MR 0342541
- Irving Kaplansky, Algebraic and analytic aspects of operator algebras, American Mathematical Society, Providence, R.I., 1970. Conference Board of the Mathematical Sciences Regional Conference Series in Mathematics, No. 1. MR 0312283
- Joachim Lambek, Lectures on rings and modules, Blaisdell Publishing Co. Ginn and Co., Waltham, Mass.-Toronto, Ont.-London, 1966. With an appendix by Ian G. Connell. MR 0206032
- Tadasi Nakayama, On Frobeniusean algebras. I, Ann. of Math. (2) 40 (1939), 611–633. MR 16, DOI https://doi.org/10.2307/1968946
- B. L. Osofsky, Rings all of whose finitely generated modules are injective, Pacific J. Math. 14 (1964), 645–650. MR 161886
- Judith D. Sally and Wolmer V. Vasconcelos, Stable rings, J. Pure Appl. Algebra 4 (1974), 319–336. MR 409430, DOI https://doi.org/10.1016/0022-4049%2874%2990012-7
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Keywords:
Uniserial ring,
projective module,
endomorphism ring
Article copyright:
© Copyright 1974
American Mathematical Society