129. Rings o f Dominant Dimension 1

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1 No. 7] Proc. Japan Acad., 44 (1968) Rings o f Dominant Dimension 1 By Toyonori KATo College of General Education, Tohoku University, Sendai (Comm. by Kenjiro SHODA, M. J. A., Sept. 12, 1968) Tachikawa [9] was the first to introduce and to study dominant dimensions for algebras. In his paper [9], dominant dimensions play a vital role in classifying QF-3 algebras. The author has found it relevant, in the study of rings with torsionless injective hulls, to introduce new dominant dimension defined by the analogue of Tachikawa [9, p. 249] using torsionless minimal injective resolutions. In Section 1, we introduce dominant dimension for torsionless modules and then give illustrative examples. In Section 2, we are concerned with rings having dominant dimension >_ 1 and show how to obtain such rings. In Theorem 1, (3), we construct a ring Q having dominant dimension oo such that QQ is a non-injective non-cogenerator in the category of right Q-modules L5JIQ. The endomorphism rings of generator-cogenerators are discussed in Section 3. Let R and Q be rings. Denote by 5WR (resp. Q) the category of right R-modules (resp. the category of right Q-modules having dominant dimension >_ 2). Then our main Theorem 2 states that Q is the endomorphism ring of a generator-cogenerator in 5iIR if and only if QQ has dominant dimension >_ 2 and there exists an equivalence SIR..EQ. In this paper, rings will have a unit element and modules will be unital. If AR is a module over a ring R, E(AR) will denote the injective hull of AR, and End(AR) the endomorphism ring of AR. We adopt the following notational trick, which will facilitate our study : homomorphisms will be written on the side opposite the scalars. 1. Dominant dimension. Let R be a ring, AR a right R-module, and let 0---~A- *X1-~X2-~... ->Xn be a minimal (essential) injective resolution of A. We shall say that A has dominant dimension >_ n if each XZ is torsionless (denoted by domi. dim. A > n). In case domi, dim. A>_ n and domi, dim. A n + 1, domi, dim. A=n. If domi. dim. A>_n for each n, domi, dim. A=oo. In case domi, dim. A 1, domi, dim. A = 0. In case R is left Artinian, torsionless injective right R-modules are always projective by Chase [3, Theorem 3.3] and hence our dominant dimension exactly coincides with that of Tachikawa [9, p. 249]. We consider some illustrative examples of dominant dimensions :

2 580 T. KATO [Vol. 44, (1) domi. dim. Z=0, where Z denotes the ring of integers. (2) domi, dim. AR>_1 if and only if E(AR) is torsionless. (3) RR is a cogenerator in (5JIR if and only if domi, dim. AR= o0 for each AR e (-511R. (4) domi, dim. RR >_ 1 if and only if domi. dim. AR >_ 1 for each torsionless AR e 51R (see Kato [5, Proposition 1]). (5) Let A=A1 e 5141, A2=E(A1)/A1, A3=E(A2)/A2f..., An =E(A1)/A1. n-n-in case domi, dim. R11>_ 1, domi, dim. A>_ n if and only if each Ai (i=1, 2,..., n) is torsionless. Proof of (5). If O--*A--3X1-->X2-~.. ->X n is an essential injective resolution of A, it can be imbedded in the following commutative exact diagram o--~a----x1--x2-~... ~Xn II l~?z l2 O->A-E(A)-E(A2)-... ->E(An) making use of the uniqueness of injective hulls. Thus (5) is straightforward in view of Kato [5, Proposition 1]. 2. A construction of rings having dominant dimension >_ n (n=1, 2, oo). Throughout this section, let R be a ring, UR a generator in 54'tR, and Q=End(UR) the ring of endomorphisms of UR. Theorem 1. Let R, UR, and Q be as above. Then (1) domi. dim. QQ >_ 1 i f and only i f E(U11) c [j UR. (2) domi. dim. QQ >_ 2 i f and only i f E(U11) C f UR and E(U11) / UR c 1 l UR. (3) There exists a ring Q o f dominant dimension oo such that QQ is a non-injective non-cogenerator,in 'IQ. Proof. We begin our proof by introducing covariant morphism functors H : 4R~-511Q and H*: 511Q->5WR as follows : H(AR)=Hom(UR, AR)Q e (511Q for each AR e 111, H*(AQ) = Hom ((Q U)Q, AQ)R e for each AQ e (5WQ where (Q U)* = RHom(Q U, QQ)Q. Note here that 0U11, 11(0U). It is just the point that Q U is finitely generated projective and R = End(Q U) (see Morita [7, Lemma 3.3]). Therefore, we have R((Q U)*0Q U)R N RHom (Q U, Q U)R N 11R11. This isomorphism establishes, for every AR e H*H(AR) = Hom ((Q U)Q, Hom (UR, AR)Q)R Hom ((Q U)*OQ UR, AR)R Hom (RR, A11)11 A11. Thus H*H is natural equivalent to the identity functor on Next, H preserves not only injectives by virtue of the projectivity of QU (for the proof, see Cartan and Eilenberg [2, Proposition 1.4, p. 107]) but also essential extensions (see Mitchell [6, Lemma 3.1, p. 88] or C. L. Walker and E. A. Walker [10, Proposition 3.2]). Therefore H

3 No. 71 Rings of Dominant Dimension? preserves injective hulls.. Furthermore, for each AR e 34, H(AR) is torsionless if and only if ARc fj UR. To see this, let ARc [-I UR. Then H(AR) c H(f UR) H H(UR) _ 11 QQ. Conversely, let H(AR) c fl QQ. Then AR H*H(AR) c H*(f QQ) N flh*(qq)= fl H*H(UR) fl UR. We are now ready to show (1), (2), and (3). (1) Since E(QQ) = E(H(UR)) = H(E(UR)), E(QQ) is torsionless if and only if E(UR) c H UR. (2) The exact sequence 0-~QQ-E(QQ)-~E(QQ) / QQ--~0 yields an exact commutative diagram 0 *H*(QQ)->H*(E(QQ))-_ H*(E(QQ) / QQ)-~0 22?Z 0-+ UR--- > E(U) ' E(UR) / UR-> 0. Thus E(UR) / UR H* (E(QQ)/ QQ). In a similar manner, we have E(QQ) / QQ c H(E(UR) / UR). The above justifies that E(QQ) c f QQ, E(QQ) / QQE fl QQ if and only if E(UR)c fl UR, E(UR) / URc fl UR. (3) Let RR be a non-injective cogenerator in 5KR (such a ring really exists by an example due to Osofsky [8, Example 2]). Let UR be infinitely generated free, and let Q=End(UR). Then QQ has dominant dimension 00. In fact, let 0- UR 3X1-*X2~... ->Xn-->... be an essential injective resolution of UR. Then O->H(UR)-~H(X1)-->H(X2)--... is also an essential injective resolution of H(UR) = QQ, for H is an exact functor and preserves injective hulls. Now observe that UR is a cogenerator in Ud LR and that Xi E fl UR for each i. Thus each H(X) is torsionless and hence QQ has dominant dimension 00. But QQ is non-injective since UR is non-injective. Moreover QQ is a non-cogenerator in (IQ. In fact, QU is a non-generator in Q(5W since UR is infinitely generated (see Morita [7, Lemma 3.3]). Therefore the trace ideal t of QU is proper, but the left annihilator of t is zero by C. L. Walker and E. A. Walker [10, Theorem 1.7]. Thus QQ is no more a cogenerator in 5ld. Remarks. (1) Let the f unctors H and H* be as above. Then H* is the left adj oint to H (see Kan [4]), for making use of the finitely generated projectivity of QU we have Horn (H*(A Q), BR) = Horn (Horn ((Q U)Q, AQ)R, BR) = Horn (Horn (Horn (Q U, QQ)Q, AQ)R, BR) Hom (Hom(QQ, AQ) Q UR, BR) Hom (AQ OQ UR, BR) Hom (A Q, Horn (UR, BR)Q) = Horn (A Q, H(BR)).

4 582 T. KATo [Vol. 44, (2) In case UR is a torsionless generator in 5R, domi. dim. QQ >_ 1 if and only if domi, dim. RR >_ 1, 3. A characterization of the endomorphism ring of a generator. cogenerator. Let Q be a ring. We denote by J the category of right Q-modules having dominant dimension >_ 2. Theorem 2. A ring Q is the endomorphism ring o f a generatorcogenerator in AlR i f and only i f QQ has dominant dimension >_ 2 and there exists an equivalence ft/tr ^' fq. Proof. Sufficiency. There exist covariant functors S : UdLR--~eCQ and T : J Q-~(±41R such that TS (resp. ST) is natural equivalent to the identity f unctor on 5lR (resp. Q) by virtue of an equivalence 514R ±Q (see Mitchell [6, Proposition 10.1, p. 61]). Put UR = T (QQ) making use of QQ e Q. Then UR is a generator in 511R since QQ is a generator in J (see Bass [1, p. 3]). Since each AQ in is torsionless, QQ is a cogenerator in f0 0 and hence UR = T (QQ) is also a cogenerator in 5118 (see Bass [1, p. 3]). Thus UR is a generator-cogenerator in 511R Moreover End (UR) = End (T(Q0)) End (QQ) Q. Necessity. Let Q = End (UR) for a generator-cogenerator UR in 5W. Then QQ has dominant dimension >_ 2 by Theorem 1, (2). Now let the f unctors H and H* be as in Proof of Theorem 1. We shall show that H gives the desired equivalence vwr ^r J'Q. The first part of the proof is to show that H has the codomain J, or equivalently, H(AR) e Q for each AR e (51IR. The next part is to show that HH*(AQ) AQ for each AQ e J. These facts together with one established in Proof of Theorem 1 will conclude the proof. Now let AR e (5WR. Then ARc fl UR since UR is a cogenerator in ±JJ R. Thus H(AR) is torsionless as was seen in the proof of the preceding theorem. Moreover the same argument as in Theorem 1 yields E(H(AR)) / H(AR) ch(e(ar) / AR) and hence E(H(AR)) / H(AR) is also torsionless. But this implies domi, dim. H(AR) >_ 2 since QQ has dominant dimension > 2 by the above. Thus H(AR) e Q for each AR E R. To show that HH*(AQ) AQ for each AQ e J, we now define a natural transformation i, ha : A0 _ HH*(AQ) = Hom (UR, Hom ((Q U)Q, AQ)R)Q by (h(a)u) f = a(u f) for a e AQ, u e UR, and f e (0U)*. In view of Proof of Theorem 1, we have HH*({j QQ) fl HH*(Q Q) fl HH*(H(UR)) ^ fj H(UR) = II QQ and this isomorphism is nothing else but the h. In case AQ is torsionless, ha is a monomorphism by the commutativity of the following diagram with an exact row

5 No. 7] Rings of Dominant Dimension >_ O-+AQ> 11 QQ 7)A 11 HH*(AQ)-jHH*(JJ QQ). Next, let 0->AQ-~ f QQ-*BQ-*O be an exact sequence with AQ injective. Notice here that BQ is torsionless since the above splits. Let us examine the following commutative exact diagram 0 --~ AQ~ fl QQ BQ--~ 0 A 22 ~1 B 0-~HH'~(AQ)-~HH*( fl QQ)->HH*(BQ)_>0. Thus 7A is an isomorphism since Y2$ is a monomorphism (using the Five Lemma). This shows that 7 A is an isomorphism in case AQ is torsionless and injective. Finally, let AQ e J. Then we have an exact sequence 0->AQ-->E(AQ)-->E(AQ)/AQ-->O with E(AQ) and E(AQ)/AQ torsionless. This sequence yields the commutative diagram with exact rows O- ~ AQ=- > E(AQ)---~E(AQ) ha l2 O_ HH*(AQ)_*HH*(E(AQ))_ HH*(E(AQ) /AQ), concluding that lia is an isomorphism. Since we have already seen that H*H(AR) AR for. each AR E 511R in Theorem 1, we have thus established the equivalence H : Remarks. (1) For any ring R 5lR has a generator-cogenerator. This observation guarantees the full generality of the above theorem. (2) Theorem 2 forms an excellent contrast with the following theorem established by Morita [7] (see also Bass [1]) : Morita Theorem. A ring Q is the endomorphism ring o f a progenerator in LIR i f and only i f there exists an equivalence SIR (3) Monomorphisms, kernels, injective hulls in J are also the corresponding ones in SJIQ. Nevertheless epimorphisms, cokernels, projectives, finitely generated objects in J Q do not necessarily coincide with the corresponding ones in 5JIQ. Added in proof. At the spring meeting of Math. Soc. Japan in 1968, by Professors K. Morita and H. Tachikawa it was announced that the results [11] related to our Theorem 2 hold for semi-primary QF-3 rings. After submitting this paper, the author received from Prof. H. Tachikawa a copy of the manuscript [12], in which he had defined the same dominant dimension as ours. He has shown by an example that the two conditions in Theorem 2 are independent for non-semi-primary Q.

6 584 T. KATO [Vol. 44, Recently Prof. K. Morita has kindly communicated to the author some results connected with Theorem 1, (1), (2). He has studied in the situation that R is right Artinian and UR is a finitely generated generator in 5W R and then has given an equivalent criterion for the ring Q = End(UR) to be of dominant dimension >_ n, introducing new self -injeetive dimension of UR. References [1] H. Bass: The Morita theorems. Lecture Notes, University of Oregon (1962). [21 H. Cartan and S. Eilenberg: Homological Algebra. Princeton University Press (1956). [31 S. U. Chase: Direct product of modules. Trans. Amer. Math. Soc., 97, (1960). [41 D. M. Kan: Adjoint functors. Trans. Amer. Math. Soc., 87, (1958). [51 T. Kato: Torsionless modules. Tohoku Math. Journ., 20, (1968). [61 B. Mitchell: Theory of Categories. Acad. Press (1965). [71 K. Morita: Duality for modules and its applications to the theory of rings with minimum condition. Sci. Rep. Tokyo Kyoiku Daigaku, Sect. A, 6, No. 150, (1958). [81 B. L. Osof sky: A generalization of quasi-frobenius rings. Journ. Algebra, 4, (1966). [91 H. Tachikawa: On dominant dimensions of QF-3 algebras. Trans. Amer. Math. Sac., 112, (1964). [10] C. L. Walker and E. A. Walker: Quotient Categories of Modules. Proceedings of the La Jolla Conference on Categorical Algebra. Springer Berlin, pp (1966). [111 K. Morita and H. Tachikawa: On semi-primary QF-3 rings (to appear). [12] H. Tachikawa: A note on QF-3 rings (to appear).