ANNALES SOCIETATIS MATHEMATICAE POLONAE Series I: COMMENTATIONES MATHEMATICAE XXI (1979) ROCZNIKI POLSKIEGO TOWARZYSTWA MATEMATYCZNEGO
Séria I: PRACE MATEMATYCZNE XXI (1979)
Ja v ed Ahsan (Islamabad, Pakistan)
Semi-local rings and quasi-projective modules
A module M over a ring Я is called quasi-injective if every homo
morphism: N -> M of any Я-submodule N of M into M can be extended to an endomorphism of M. Dually, M is called quasi-projective if for every epimorphism (p: M -> N and every homomorphism g: M -> N , N being an Я-module, there exists an endomorphism f of M such that ( pof = g. M will be called semi-simple if the Jacobson-radical of M is zero. A ring Я is called local if R/J is a division ring: J denotes the Jacobson radical of Я.
Я is called semi-local if case R/J is semi-simple artinian. Semi-simple artinian rings themselves have been characterized homologically in a number of different ways. For example we may recall the well-known fact that rings all of whose modules are injective are semi-simple artinian. Osofsky [7]
proved that rings whose cyclic modules are injective are semi-simple artinian.
In [3], Cateforis and Sandomierski characterized commutative semi-simple artinian rings as those rings whose semi-simple cyclic modules are injective.
Michler and Villamayor [6] later proved that this characterization remains valid in the general case also. As for rings with similar conditions in the projective setting it is enough to recall that rings over which every simple module is projective are semi-simple artinian (see Satyanarayana [9]).
By relaxing the conditions of injectivity (projectivity) on modules to quasi
injectivity (quasi-projectivity), larger classes of rings have been studied in a number of subsequent papers. To start with, we may note the well-known fact that rings all of whose modules are quasi-injective (quasi-projective) are semi-simple artinian. In fact, this character can be obtained by just assuming that finitely generated modules are quasi-injective (quasi-projective). The proof in the quasi-injective case given in Proposition 1 below and the quasi- projective case, though known already, can be proved analogously. However, rings all whose cyclic modules are quasi-injective (quasi-projective) need not even be semi-simple. For example, any proper factor ring of the ring of integers has these properties without being semi-simple. Rings whose cyclic piodules are quasi-injective (quasi-projective) have been called qc (q*) rings
6 J. Ahsan
and investigated by Ahsan [1] and Koehler [5] (Koehler [4]). Moving in this direction further, it can be observed that if Я is a local ring, then each semi-simple finitely generated Я-module is quasi-injective (quasi-projective).
However, it is not necessary for a local ring to be qc (<?*). Inspired by this example, the author, in an earlier paper [2], studied rings all of whose semi-simple finitely generated modules are quasi-injective. It was proved that this property characterizes semi-local rings. The purpose of the present brief note is to study rings all of whose semi-simple finitely generated modules are quasi-projective. It will be proved that this property also characterizes semi-local rings. Throughout this note we shall assume that all rings have the identity element and all modules are unitary right modules. We now start with the following proposition.
Proposition I. Let R be any ring. Then the following statements are equivalent:
1° Each finitely generated R-module is quasi-injective.
2° R is semi-simple artinian.
P roof. Let M R be any cyclic Я-module. Consider AR = Я © М ; then Ar is a finitely generated Я-module and so quasi-injective. Since Rr Ar
and Ar is quasi-injective, any map f : I - + A R; I any right ideal of Я;
extends to a map f : Rr -> AR. Then by Baer’s Criterion, AR is injective.
Therefore M R is injective. Hence by the results of Osofsky [7], Я is semi
simple artinian.
Lemma 2. Let R be any ring. I f each semi-simple finitely generated R-module is quasi-projective, then every factor ring of R has this property.
P roof. Let Я be a factor ring of Я. Suppose M R is a semi-simple finitely generated Я-module. Then M R is also semi-simple and finitely gen
erated. Hence by the assumption M R is Я-quasi-projective. Then M R is Я-quasi-projective.
Since any epimorphism from a semi-simple artinian module onto an Я-module splits, the following lemma is immediate.
Lemma 3. Each semi-simple artinian module is quasi-projective.
The next lemma is analogus to the lemma in Rangaswamy and Vaneja [8].
Lemma 4. Let A ® В be a quasi-projective R-module. Then every epi
morphism from A to В splits.
We are now ready to prove the following proposition.
Proposition 5. Let R be any ring. Then the following statements are equivalent :
(1) Each semi-simple finitely generated R-module is quasi-projective.
(2) Я is a semi-local ring.
P roof. 1° (1)=>(2). Let Я = Я/J: / denotes the Jacobson-radical of Я.
Semi-local rings and quasi-projective modules 1
Then in view of Lemma 2 above, each semi-simple finitely generated Я-module is quasi-projective. Let Mr be any simple Я-module. Consider Ar = Rr® Mr. Since Ar is a semi-simple finitely generated Я-module, Ar
is quasi-projective. Then in view of Lemma 4, above, the epimorphism:
Rr -> M R splits and so M R is projective. Since every simple Я-module is projective, Я is semi-simple artinian. This implies that R is a semi-local ring.
2° (2) => (1). Let us now suppose that R is a semi-local ring. In order to prove that each semi-simple finitely generated Я-module is quasi-projective, we proceed using an indictive argument. Let M R be any semi-simple Я-module generated by n elements. If n = 1, then M R is a cyclic module. This means that M R % R/ I : for some right ideal I of R. Since J (R/I) = 0, J ç= / , so I/J ç R/J (as submodules). Then R/I % R/J/l/J. Since Я is a semi-local ring, RjJ is semi-simple artinian and so R/I, being a homomorphic image of a semi-simple artinian module, is semi-simple artinian. Suppose now that each semi-simple Я-module generated by n = к elements is semi-simple artinian. Let M R = R x t -1- ... + R x k +l . Call AR = R x t and consider the short exact sequence:
0 -+ A -» M -► M/ A -► 0.
The fact that A and M/ A are artinian implies that M is artinian. Thus we have proved that every semi-simple finitely generated Я-module is semi
simple artinian and hence in view of Lemma 3 above, each such module is quasi-projective. This completes the proof of the proposition.
References
[1] J. A hsan, Rings all of whose cyclic modules are quasi-injective, Proc. London Math.
Soc. 27 (3) (1973), p. 425-439.
[2] — A characterization of semi-local rings (to appear).
[3] V. C. C a te fo r is and F. L. S a n d o m ie rs k i, On commutative rings over which the singular submodule is a direct summand for every module, Pacific J. Math. 31 (1969), p. 289-292.
[4] A. K o e h le r, Rings for which every cyclic module is quasi-projective, Math. Ann. 189 (1970), p. 311-316.
[5] — Rings with quasi-injective cyclic modules, Quart. J. Math. Oxford, Sér. 2 (1974), p. 51-55.
[6] G. O. M ic h le r and О. E. V illa m a y o r, On rings whose simple modules are injective, J. Algebra 25 (1973), p. 185-201.
[7] B. L. O sofsky, Non-injective cyclic modules, Proc. Amer. Math. Soc. 19 (1968), p. 1383-1384.
[8] K. R a n g a sw a m y and N. V an eja, Quasi-projectives in abelian and module categories, Pacific J. Math. 43 (1972), p. 221-238.
[9] M. S a ty a n a r a y a n a , Semi-simple rings, Amer. Math. Monthly 74 (1967), p. 1086.
DEPARTMENT OF MATHEMATICS CALIFORNIA STATE UNIVERSITY HAYWARD, CALIFORNIA DEPARTMENT OF MATHEMATICS UNIVERSITY OF ISLAMABAD ISLAMABAD, PAKISTAN
