ON COLOCAL PAIRS

Abstract

In [9, Theorem 3.1] K. R. Fuller characterized indecomposable injective projective modules over artinian rings using i-pairs. In [3] the author generalized this theorem to indecomposable projective quasi-injective modules and indecomposable quasiprojective injective modules over artiniain rings. In [2] the author and K. Oshiro studied the above Fuller’s theorem minutely. Further in [12] M. Hoshino and T. Sumioka extended these results to perfect rings. In this paper we studies the results in [3] from the point of view of [2], [12]. 1. ON FULLER’S THEOREM AND PAST RESULTS Throughout this paper, we let R be a semiperfect ring. By MR (resp. RM) we stress that M is a unitary right (resp. left) R-module. For an R-module M , we denote the injective hull, the Jacobson radical, the socle, the top M/J(M), and the composition length of M by E(M), J(M), S(M), T (M), and |M |, respectively. Further we denote the right ( resp. left ) annihilator of T in S by rS(T ) ( resp. lS(T ) ). Definition 1. Let M, N be R-modules. We say that M is N-injective if, for any submodule X of N and any R-homomorphism φ : X → M , there exists φ : N → M with φ|X = φ. And we say that M is N -simple-injective if, for any submodule X of N and any R-homomorphism φ : X → M with Imφ simple, there exists φ : N → M with φ|X = φ. Definition 2. Let e, f be primitive idempotents in R and let g be an idempotent in R. We say that R satisfies αr[e, g, f ] if rgRf leRg(X) = X for any right fRf -submodule X of gRf with rgRf (eRg) ⊆ X. And we say that R satisfies αl[e, g, f ] if leRgrgRf (Y ) = Y for any left eRe-submodule Y of eRg with leRg(gRf) ⊆ Y . Further we say that (eR,Rf) is an injective pair ( abbreviated i-pair ) if S(eRR) ∼= T (fRR) and S(RRf) ∼= T (RRe). The following theorem is given by K. R. Fuller in [9]. By this theorem, indecomposable projective injective right R-modules over right artinian rings are characterized using ipairs. Theorem 3. (Fuller ) Let R be a right artinian ring and let e, f be primitive idempotents in R. Then the following are equivalent. The detailed version of this paper will be submitted for publication elsewhere. (a) eRR is injective with S(eRR) ∼= T (fRR). (b) (eR,Rf) is an i-pair. (c) R satisfies αr[e, 1, f ] and αl[e, 1, f ]. In [2] Theorem 3 is minutely studied by the author and K. Oshiro over semiprimary rings as follows. Theorem 4. (Baba, Oshiro ) Let R be a semiprimary ring and let e, f be primitive idempotents in R. (I) The following are equivalent. (a) eRR is injective. (b) (i) There exists a primitive idempotent f in R with (eR,Rf) an i-pair. (ii) R satisfies αr[e, 1, f ]. (II) Suppose that (eR,Rf) is an i-pair. (1) If ACC holds on right annihilator ideals, then (i) αr[e, 1, f ] holds, (ii) the equivalent conditions in the following (2) hold. (2) The following are equivalent. (a) | eReeR | < ∞. (b) |RffRf | < ∞. (c) Both eRR and RRf are injective. Theorem 4 is further considered over perfect rings by M. Hoshino and T. Sumioka in [12]. And the following theorem is given. Theorem 5. (Hoshino, Sumioka ) Let R be a left perfect ring and let e, f be primitive idempotents in R. (I) The following are equivalent. (a) eRR is R-simple-injective. (b) There exists a primitive idempotent f in R such that (i) (eR,Rf) is an i-pair. (ii) R satisfies αr[e, 1, f ]. (II) Suppose that (eR,Rf) is an i-pair. Then the following are equivalent. (a) | eReeR | < ∞. (b) |RffRf | < ∞. (c) Both eRR and RRf are injective. On the other hand, in [3] the author generalized Theorem 3 to indecomposable projective quasi-injective modules and indecomposable quasi-projective injective modules over artiniain rings as follows. Theorem 6. (Baba ) Let R be a semiprimary ring and let e, f be primitive idempotents in R. Suppose that DCC holds on { rRf (I) | eReI ⊆ eR }. Then the following are equivalent. (a) eRR is quasi-injective with S(eRR) ∼= T (fRR). –2– (b) E(T (RRe)) is quasi-projective of the form RRf/rRf (eR). (c) S(eRR) ∼= T (fRR) and S(eReeRf) is simple.