A surprising amount of information about right self-injective rings can be obtained by studying a much larger class of rings, the right mininjective rings. A ring is called right mininjective if every isomorphism between two simple right ideals is given by left multiplication. The basic general facts about these rings are derived in this chapter and this work serves as a basis for the study of two important subclasses: the P-injective rings in Chapter 5 and the simple injective rings in Chapter 6.

After giving several examples (including a right mininjective ring that is not left mininjective), we show that mininjectivity is a Morita invariant and that “min” versions of the C2- and C3-conditions hold. (The “min” version of the C1-condition is studied in Chapter 4.) Surprisingly, under a mild commutativity condition, a ring R is right minininjective if and only if its right socle is square-free, and every factor ring of R is right mininjective if and only if R has a distributive lattice of right ideals. In general, it is shown that the right socle of any right mininjective ring R is contained in the left socle (in fact, if kR is a simple right ideal then Rk is also simple). This remarkable fact is used repeatedly throughout the book. If a right mininjective ring is semiregular and has essential right socle, we show that the right singular ideal equals the Jacobson radical, extending the situation for right self-injective rings.

In this chapter, we consider the class of left min-CS rings (for which every minimal left ideal is essential in a direct summand) and show that this weak injectivity property is useful in obtaining semiperfect rings. Indeed, it is proved in Theorem 4.8 that if R is left min-CS, then the dual of every simple right R-module is simple, if and only if R is semiperfect with Sι = Sr and soc(Re) is simple and essential for every local idempotent e of R. The hypotheses of Theorem 4.8 are the weakest known conditions of this type that imply that R is semiperfect.

If we strengthen the left min-CS hypothesis in Theorem 4.8 by requiring that each closed left ideal with simple essential socle be a direct summand of RR (R is left strongly min-CS), we obtain a class of rings that satisfies many of the characteristic properties of left PF rings. If instead of assuming in Theorem 4.8 that the duals of simple right R-modules are simple we suppose, more generally, that R is right Kasch, then we obtain a larger class of rings that still retains many of these properties: It is shown in Theorem 4.10 that R is left CS and right Kasch if and only if it is semiperfect and left continuous with Sr ⊆essRR.

We now introduce another class of right mininjective rings that seems to effectively mirror the properties of right self-injective rings. A ring R is called right simple injective if every R-linear map with simple image from a right ideal to R extends to R. Examples include right mininjective rings and right self-injective rings, but there are simple injective rings that are not right P-injective, and there exist right simple injective rings that are not left simple injective. If R is right simple injective, so also is eRe, where e is an idempotent satisfying ReR = R, but simple injectivity is not a Morita invariant [we characterize when Mn(R) is right simple injective].

We show that a simple injective ring R is right Kasch if and only if every right ideal is an annihilator and, in this case, that R is left P-injective, soc(eR) is simple and essential in eR for every local idempotent e in R, and R is left finite dimensional if and only if it is semilocal. In fact, if R is semiperfect, right simple injective and soc(eR) ≠ 0 for every local idempotent e (the analogue of the right minfull rings), then R is a right and left Kasch, right and left finitely cogenerated, right continuous ring in which Sr = Sl (= S) and both soc(eR) = eS and soc(Re) = Se are simple and essential in eR and Re, respectively, for every local idempotent e in R.

A ring R is right mininjective if R-linear maps aR → R extend to RR → RR whenever aR is a simple right ideal. It is natural to enquire about the rings for which this condition is satisfied for all principal right ideals aR (for example if R is regular or right self-injective). These rings, called right principally injective (or right P-injective), play a central role in injectivity theory. An example is given of a right P-injective ring that is not left P-injective. If R is right P-injective it is shown that Zr = J, that R is directly finite if and only if monomorphisms RR → RR are epic, and that R has the ACC on right annihilators if and only if it is left artinian. If R is right P-injective and right Kasch then Sι ⊆ essRR. A semiperfect, right P-injective ring R in which Sr ⊆ess RR is called a right GPF ring. Hence the right PF rings are precisely the right self-injective, right GPF rings, and these right GPF rings exhibit many of the properties of the right PF rings: They admit a Nakayama permutation, they are right and left kasch, they are left finitely cogenerated, Sr = Sι is essential on both sides, and Zr = J = Zι.

Unlike mininjectivity, being right P-injective is not a Morita invariant. In fact, Mn(R) is right P-injective implies that R is right n-injective.

A theorem of Faith and Walker asserts that a ring R is quasi-Frobenius if and only if every injective right R-module is projective and hence that every right module over a quasi-Frobenius ring embeds in a free module. There is an open problem here. If we call a ring R a right FGF ring if every finitely generated right R-module can be embedded in a free right R-module, it is not known if the following assertion is true:

The FGF-Conjecture. Every right FGF ring is quasi-Frobenius

Here are four important results on the conjecture:

1. Every left Kasch, right FGF ring is quasi-Frobenius.

2. Every right self-injective, right FGF ring is quasi-Frobenius.

3. Every right perfect, right FGF ring is quasi-Frobenius.

4. Every right CS, right FGF ring is quasi-Frobenius.

We prove all these assertions; in fact we capture all of (1), (2), and (3) in Theorem 7.19: If Mn(R) is a right C2 ring for each n ≥ 1 and every 2-generated right R-module embeds in a free module then R is quasi-Frobenius. This theorem also implies that the FGF-conjecture is true for right FP-injective rings, and it reformulates the conjecture by showing that it suffices to prove that every right FGF ring is a right C2 ring. Furthermore, the theorem shows that the conjecture is true for semiregular rings with Zr = J. We call these rings right weakly continuous, and investigate their basic properties.

The structure of a semisimple artinian ring is well known thanks to the Wedderburn–Artin theorem. Moreover, if R is right artinian then J is nilpotent and R/J is semisimple, so a major problem is to “lift” the structure of R/J to R. This leads to lifting idempotents and hence to the notion of a semiperfect ring. These rings are described in this section, together with a discussion of the important subclass of right perfect rings. Finally, the situation where R/J is regular is treated under the heading of semiregular rings. Each of these classes of rings is used extensively throughout the book.

Semiperfect Rings

Semiperfect rings are perhaps the most useful generaliz ation of the classical artinian rings. We begin by describing the simplest examples.

Proposition B.1.The following conditions are equivalent for a ring R:

1. R/J is a division ring.

2. R – J consists of units.

3. If a є R then either a or 1 – a is a unit.

4. R has a unique maximal right (respectively left) ideal.

5. J is a maximal right (respectively left) ideal.

Proof. (1)⇒(2) becausea is a unit in R whenever a + J is a unit in R/J, (2)⇒(3)⇒(4)⇒(5) are clear, and (5)⇒(1) because aR = R whenever a ∉ J by (5) because J ⊆sm RR.

A ring R is called local if it satisfies the conditions in Proposition B.1, and an idempotent e in R is called a local idempotent if eRe is a local ring. The following result will be used repeatedly.

The Faith conjecture asserts that every left or right perfect, right self-injective ring R is quasi-Frobenius. The conjecture remains open for semiprimary, local, right self-injective rings with J3 = 0. It is known (Theorem 3.40) that the conjecture is true if J2 = 0. In this section we construct a local ring R with J3 = 0 and characterize when R is artinian or self-injective in terms of conditions on a bilinear mapping from a (D, D)-bimodule to a division ring D ≅ R/J. We conclude by characterizing other properties of R in a similar way.

Generalities

If S is any ring and SVS, SWS, and SPS are bimodules, a function V × W → P, which we write multiplicatively as (υ,ω) ↦ υω, will be called a bimap if the conditions

(1) (υ + υ1)ω = υω + υ1ω and (Sυ)ω = S(υω),

(2) υ(ω + ω1) = υω + υω1 and υ(ωS) = (υω)S, and

(3) (υS)ω = υ(Sω)

hold for all υ, υ1 in V, all ω, ω1 in W, and all s in S. Our interest is in the case when S = D is a division ring. We construct the following ring, which is the central topic of this chapter.

Definition. Let DVD and DPD be nonzero bimodules over a division ring D, and suppose a bimap V × V → P is given.

A ring R is called quasi-Frobenius if it is right or left self-injective and right or left artinian (all four combinations being equivalent). The study of these rings grew out of the theory of representations of a finite group as a group of matrices over a field – the corresponding group algebra is quasi-Frobenius. At the turn of the twentieth century G. Frobenius carried out fundamental work on representations of “hypercomplex systems” – finite dimensional algebras in modern terminology. This topic was revied in the late 1930s and early 1940s by Brauer, Nesbitt, Nakayama, and others in their study of “Frobenius algebras.”. Nakayama introduced quasi-Frobenius rings in 1939 and, in 1951 Ikeda characterized them as the left and right self-injective, left and right artinian rings. The subject is intimately related to duality, the duality from right to left modules induced by the hom functor, and, more importantly for us, the duality related to annihilators. The present extent of the theory is vast, and we make no attempt to be encyclopedic here. Instead we provide an elementary, self-contained account of the basic facts about these rings at a level allowing researchers and graduate students to gain entry to the field. This pays off by giving new insights into some of the outstanding open questions about quasi-Frobenius rings.

Our approach begins by extending many earlier results to a much wider class of rings than heretofore investigated. We call these rings mininjective.

A ring R is called a right Johns ring if it is right noetherian and every right ideal is an annihilator. All these rings were once thought to be right artinian, but in 1992 Faith and Menal gave an example of a right Johns ring that is neither right nor left artinian (see Example 8.16 in Section 8.3). In 1993, Faith and Menal asked whether every strongly right Johns ring is quasi-Frobenius, where R is called a strongly right Johns ring if Mn(R) is a right Johns ring for all n ≥ 1, equivalently if R is right noetherian and left FP-injective (by Theorem 5.41 since right Johns rings are left P-injective). This question remains open. Note that Björk's example (Example 2.5) is a left and right artinian ring R that is left Johns but not quasi-Frobenius. Hence M2(R) cannot be left Johns since, otherwise, R would be a left 2-injective ring by Proposition 5.36, which is a contradiction. Thus R is a two-sided artinian, left Johns ring that is not strongly left Johns.

The Faith–Menal example has the additional property that the right socle is essential as a right and as a left ideal. Hence it shows that the following theorem of Ginn and Moss does not have a one-sided version: A left and right noetherian ring with essential right socle is left and right artinian.

Only in the semiperfect case does the full power of the mininjective hypothesis comes into focus. The reason is that we can get combinatorial information about the simple right and left ideals because a semiperfect ring has only a finite number of simple right modules that are represented in the ring by basic idempotents. We remind the reader that all the facts needed about semiperfect rings are developed in Appendix B.

We begin by characterizing the semiperfect, right mininjective rings, and we use the result to give easy proofs of two well-known characterizations of quasi-Frobenius rings: R is quasi-Frobenius if and only if it is left and right artinian, both soc(Re) and soc(eR) are simple for every local idempotent e in R, and either Sr = Sι or R is right and left Kasch. The following class of rings is important in extending these results. A Ring R is called a right minfull ring if it is semiperfect, right mininjective, and soc(eR) ≠ 0 for every local idempotent e ∈ R. We show that these right minfull rings are a Morita invariant class that exhibits many of the basic properties of quasi-Frobenius rings. In particular, they are right and left Kasch, and Sr = Sι if and only if lr(K) = K for every minimal left ideal K that is contained in Re for some local idempotent e.