Rings whose cyclics are essentially embeddable in projective modules. (English) Zbl 0698.16012

Jain, S. K.; López-Permouth, S. R.

J. Algebra 128, No. 1, 257-269 (1990).

The authors call a ring \(R\) a right (left) CEP-ring if each cyclic right (left) \(R\)-module is essentially embeddable in a projective module. Quasi-Frobenius (QF) rings and right uniserial rings are examples of right CEP-rings and in fact a theorem of C. Faith and E. A. Walker [J. Algebra 5, 203-221 (1967; Zbl 0173.03203)] shows that \(R\) is QF if and only if \(R\) is both a right and a left CEP-ring. Here it is shown that if \(R\) is both a QF-3 ring and a right CEP-ring then \(R\) is QF. (Recall that \(R\) is said to be QF-3 if its right injective hull is projective.) Moreover it is shown that a semiperfect ring \(R\) is right CEP if and only if (i) \(R\) is right Artinian and (ii) every indecomposable projective module is both uniform and weakly \(R\)-injective. (Here a module \(P\) is called weakly \(R\)-injective if, given any \(f\in\operatorname{Hom}(R,E(P))\), where \(E(P)\) is the injective hull of \(P\), then \(f(1)\) belongs to some submodule \(X\) of \(E(P)\) isomorphic to \(P\). Also, as a “Note added in proof”, the authors say that the uniform assumption may be omitted in this characterization.) Another main result characterizes those rings \(R\) of which every homomorphic image is a right CEP-ring as being a direct sum of right uniserial rings or matrix rings over right self-injective right uniserial rings. The paper ends with illustrative examples obtained using the trivial (split) extension construction.

Reviewer: J.Clark

MSC:

16L60	Quasi-Frobenius rings
16L30	Noncommutative local and semilocal rings, perfect rings
16D50	Injective modules, self-injective associative rings
16D40	Free, projective, and flat modules and ideals in associative algebras
16D70	Structure and classification for modules, bimodules and ideals (except as in 16Gxx), direct sum decomposition and cancellation in associative algebras)

Keywords:

trivial extension; quasi-Frobenius rings; right uniserial rings; right CEP-rings; left CEP-ring; QF-3 ring; right injective hull; semiperfect ring; indecomposable projective module; direct sum of right uniserial rings; matrix rings over right self-injective right uniserial rings

Citations:

Zbl 0173.03203

References:

[1]	Anderson, F. W.; Fuller, K. R., Rings and Categories of Modules (1974), Springer-Verlag: Springer-Verlag New York/Heidelberg/Berlin · Zbl 0242.16025
[2]	Faith, C., Algebra: Rings, Modules and Categories I (1973), Springer-Verlag: Springer-Verlag New York/Berlin/Heidelberg · Zbl 0266.16001
[3]	Faith, C., Algebra: Rings, Modules and Categories II (1976), Springer-Verlag: Springer-Verlag New York/Berlin/Heidelberg
[4]	Faith, C., Self-injective rings, (Proc. Amer. Math. Soc., 77 (1979)), 157-164 · Zbl 0424.16014
[5]	Faith, C., Embedding modules in projectives: A report on a problem, (Advances in Non-commutative Ring Theory. Advances in Non-commutative Ring Theory, Lectures Notes in Mathematics, Vol. 951 (1981), Springer-Verlag: Springer-Verlag New York/Berlin/Heidelberg) · Zbl 0504.16009
[6]	Faith, C.; Walker, E. A., Direct sum representation of injective modules, J. Algebras, 5, 203-221 (1967) · Zbl 0173.03203
[7]	Jain, S. K.; López-Permouth, S. R., A generalization of the Wedderburn-Artin theorem, (Proc. Amer. Math. Soc., 106 (1987)), 19-23 · Zbl 0681.16016