Manabu Harada

On extending property on direct sums of uniform modules

of E by indecomposable modules Ea such that E=^@EΛ and the socle of EΛ is Sa. We shall call the first property and the second propert the extending property of simple module and of decomposition, respectively. These concepts are dual to those of lifting properties mentioned in Γ7]. We shall study the above properties on direct sums oί completely indecomposable modules with certain condition over an arbitiary ring. We shall give characterizations of those properties in terms of endomorphisms over direct summands and show that quasi-injective modules and generalized uniserial rings [15] are related to those properties. Our results are dual or similar to those in [9] and are applied to the study of QF-2 rings in [8].

On endomorphism rings of noetherian quasi-injective modules

Let R be a ring with identity element. One of the authors studied the endomorphism ring of projective right i?-module P with chain conditions in [6] and showed that the ring is right artinian (resp. noetherian) if so is P as an i?-module. We shall consider its dual in this short note. Unfortunately, we could not give the complete dual of them. Recently, many authors have studied structures of injective module Q and given many interesting results between ideals in R and S-submodules in Q, where S=HomR(Q, Q). However, we shall study mainly, in this note, some properties between i?-submodules and left ideals in S. In the first section, we shall consider the above problem in an abelian C3category A (see [10], Chap. Ill), and show that if-A is a quasi-injective object in A and A is noetherian (resp. artinian), then the endomorphism ring [A, A] of A is semi-primary (resp. left noetherian). In the second section, we shall study conditions under which *S— HomR(My M) is left artinian, when M is a right i?-quasi-injective noetherian module and shall give a condition that M gives us a Morita duality on categories of finitely generated right i?-(resp. left /S)-modules. In this paper, we always assume that i?-modules M are unitary and the ring of endomorphism of M operates from the left side. After having completely settled this note, we have found J.W. Fishers results in [5]. His Theorem 2 is contained in [6], Theorem 2. 8 and Theorem 3 coincides with our Theorem 1. Further, K. Motose obtained similar results in [12].

QF-3 and Semi-Primary PP-Rings. I

Recently the author has given a characterization of semi-primary hereditary ring in [4]. Furthermore, those results in [4] have been extended to a semi-primary PP-ring in £3], (a ring A is called a left PP-ring if every principal left ideal in A is A-projective}. This short note is a continuous work of [3] and [4]. Let K be a field and A an algebra over K with finite dimension. A is called a QF-3 algebra if A has a unique minimal faithful representation ([10~1). Mochizuki has considered a hereditary QF-3 algebra in [6]. In this note we shall study a PP-ring with minimal condition or of semi-primary. To this purpose we generalize a notion of QF-3 algebra in a case of ring. We call A left (resp. right} QF-3 ring if A has a faithful, injective, projective left (resp. right} ideal, (cf. [5], Theorems 3.1 and 3. 2). Let 1 = Σ ^ * be a decomposition of the identity element 1 of a semi-primary ring A into a sum of mutually orthogonal idempotents such that Ef modulo the radical N is the identity element of simple component of A/N. If Ax is A-projective for all x^EfAEjy we call A a partially PP-ring, (see [3], §2). Such a class of rings contains properly classes of semi-primary hereditary rings and PP-rings. Our main theorems are as follows : Let A be directly indecomposable and a left QF-3 ring and semi-primary partially PP-ring. Then 1) there exists a unique primitive idempotent e in A (up to isomorphism) such that eN = (0) and every indecomposable left injective ideal in A is faithful, projective and isomorphic to Ae. Furthermore, A is a right QF-3 ring. 2) Let B = HomeAe(Ae, Ae\ where Ae is regarded as a right eAe-module. Then eAe is a division ring and B = (eAe)n \ B is a left and right injective envelope of A as an A-module and B is A-projective. Furthermore, if A is hereditary, then A is a generalized uniserial ring whose basic ring is of triangular matrices over a division ring. (Mochizuki proved in [6] the above fact 2) in a case of hereditary algebra over a field with finite dimension).

Almost QF rings and almost QF# rings

In this paper we assume that every ring R is an associative ring with identity and R is two-sided artinian. The author has defined almost projective modules and almost injective modules in [8], and by making use of the concept of almost projectives he has defined almost hereditary rings in [7], whose class contains that of hereditary rings and serial rings. Similarly to [7] we shall define an almost QF ring, which is a generalization of QF rings. It is well known that an artinian ring R is QF if and only if R is self injective. Following this fact, if R is almost injective as a right J?-module, we call R a right almost QF ring. Analogously we call R a right almost QF* ring if every injective is right almost projective. On the other hand, the author studied rings with (*) (resp. (*)*) (see §1 for definitions) in [4]. K. Oshiro called such a ring a right H(resp. co-H) ring in [10]. In this note we shall show that a right almost QF (resp. almost QF) ring coincides with a right co-H (resp. H-) ring. In the final section we shall give a characterization of serial rings in terms of almost projectives and almost injectives. In the forthcoming paper [9] we shall study certain conditions under which right almost QF rings are QF or serial.

On almost relative injectives on Artinian modules

We have introduced a concept of almost relative projectives (resp. injectives) in [7] (resp. [2]) which is deeply related with lifting modules [9] (resp. extending modules [10]). When we study further those modules, we have understood that it is necessary to generalize [2], Theorem to a case of artinian modules. Namely, we shall give the following theorem (Theorem 2): let U and {[/,-, //}?.i y-i be LE and artinian modules such that U is 7,-injective for ally and U is almost £/rinjective but not Ϊ7rinjective for all i. Then U is almost if and only if Σ , θ ^ i is an extending module.

Almost M-projectives and nakayama rings☆

In 1960, H. Bass [3] defined the notion of semi-perfect rings and, in 1963, E. Mares [l l] generalized it to that of semi-perfect modules. Since that time various kinds of further generalizations have been considered. In particular, K. Oshiro [ 161 studied quasi-semiperfect modules in connection with several important conditions on direct summands. The present paper is closely related to his paper as well as to the papers [ 12, 151 by Mueller and others and is cncerned with the condition (D,) (which is (C,) in [16]) over modules with direct decomposition into hollow modules. This is indeed motivated by the theorem [ 16, Theorem 3.51 that every quasisemiperfect module has the above type of decomposition. In Section 2 we shall clarify relationships between M-projectives over modules with direct decomposition into hollow modules and the condition (D,), which is dual to Cl.5, Theorem 81. In order to translate (D, ) in terms of homomorphisms, we shall introduce a new concept of almost A4-projectives to find some relations between almost M-projectives and a weaker condition (D’,). After preparing several basic results in Section 3, we shall, in Section 4, restrict ourselves to semi-perfect rings and give a series of characterizations of right Nakayama rings with special properties by making use of the concept of almost M-projectives. Finally, we shall classify modules with (D,) over a local Dedekind domain in Section 5.

Generalizations of Nakayama ring. III

In the previous papers [6] and [7], we gave several rings generalized from Nakayama rings. We shall study the same problem, in this note, following those methods. In the first two sections, we shall consider some right artinian rings with properties (*, 1) and (*, 2), respectively (see §1), and we shall give the complete types of US-4 algebras with J=0 over an algebraically closed field in the third section. In the final section, we shall give a structure of US-4 algebras with (*, 1).

On maxi-quasiprojective modules

We have defined a mini-injective module and given some structures of self mini-injective rings and certain relationships between such rings and QF-rings in [8] and [9]. In this short note we shall study the modules dual to mini-injective modules, which we call maxi-quasiprojective modules. We shall give a characterization and some structures, in terms of the above modules, of those rings whose every injective modules has the lifting property of direct decompositions modulo the Jacobian radical (see [5], [6] and [7]). Furthermore, we shall show that the above rings are closely related to QF-rings (see [8] and [9]).

Artinian rings related to relative almost projectivity. II

Let R be an artinian ring. In [10], we have studied R on which the following condition holds : for i?-modules M and N, if M is iV-projective, then M is always almost iV-projective for every submodule M of M. If M is always Af-projective in the above, then this property characterizes hereditary rings with / 2 = 0 [2] and [6], where / is the Jacobson radical of R. We have investigated the above condition in [10], when i) M and N are local and ii): M is local and N is a direct sum of local modules. In this paper we give a characterization of R over which the above condition is satisfied for any i?-modules M and N.

Simple Submodules in a Finite Direct Sum of Uniform Modules

Let R be a ring with identity. We shall study submodules in a direct sum of uniform modules Ui with finite length. Let Si be the socle of Ui. Then we obtain the natural imbedding of EndR (Ui) / J (EndR (Ui)) into EndR (Si), where J(EndR (Ui)) is the Jacobson radical of End (Ui), We shall denote them by ⊲ (Ui), ⊲, respectively.