On ps-Supplement Submodules

Let be given two rings R and T, a bimodule with essential socle on both sides and such that U-duals of simples are of finite length, a left R-module , a right T-module and a non degenerate bilinear map ; we show that if either or satisfies both the A.C.C. and the D.C.C. on annihilators with respect to the map , then and are both of finite length. As an application, we generalize a theorem of Faith on finitely embedded commutative rings. Other situations in which the result is applicable are illustrated by a classic theorem of T. Lenagan.

This thesis studies the conditions under which certain U (G)-modules have integral dimension. Specifically, the paper explores the circumstances under which certain U (G)-modules are free modules. Two main theorems are proven for this endeavor. The first theorem takes the matrix representation A of a submodule N of a free left module M over an arbitrary ring R. The theorem states that if A can be put into reduced row echelon form, then N and M/N are both free left R-modules. The second theorem takes the matrix representation B of an arbitrary left R-module homomorphism T. If B can be put into reduced column echelon form, then Ker( T) is a free left R-module as well. The results of these theorems are well-known if R is a field or division ring. The results are also probably known for an arbitrary ring R , but there does not seem to be a readily accessible source in the standard literature.

The condition that certain left ideals in a finite monoid generate projective left ideals in the semigroup algebra imposes a strong restriction on the intersection of principal left ideals in the semigroup. Let S be a finite monoid, k be a commutative ring with identity, and let Ic S be a left ideal in S. We demand that kI be projective as a left kSmodule and investigate the resulting restrictions on the structure of I. In particular we can look for necessary conditions on S for kS to be left hereditary. (A sufficient condition is obtained in [4] and [5].) Semigroup terminology below follows [1] and [2]. We first need the following facts, which are valid in any ring with identity. LEMMA 1. Let R be a ring with identity. Let Ic R be a left ideal which is projective as a left R-module. Let e E R be any idempotent. Then the left ideal I+ Re is projective if and only if JrflRe is a direct summand of L PROOF. We observe that we have the following two short exact sequences: 0 I(1 e) -* I + Re Re > 0, 0OI n Re-I--I I(1 -e)->*O, where the map on the right end of the first sequence is xH-*xe, which has kernel (I+Re) r)R(1 -e)=I(1 -e), and the map on the right end of the second sequence is xF-*x(1 -e). Since Re is projective, the first sequence always splits, so that I+Re is projective if and only if I(1 -e) is. On the other hand, since I is projective, 1(1 -e) is projective if and only if the second sequence splits. Received by the editors June 5, 1972. AMS (MOS) subject classifications (1970). Primary 20M10, 20M25; Secondary 16A32, 16A50, 16A60.

A module is uniserial in case its submodules are linearly ordered by inclusion. A ring R is left (right) serial if it is a direct sum of uniserial left (right) R-modules. A ring R is serial if it is both left and right serial. It is well known that for artinian rings the property of being serial is equivalent to the finitely generated modules being a direct sum of uniserial modules [8]. Results along this line have been generalized to more arbitrary rings [6], [13].This article is concerned with investigating rings whose indecomposable injective modules are uniserial. The following question is considered which was first posed in [4]. If an artinian ring R has all indecomposable injective modules uniserial, does this imply that R is serial? The answer is yes if R is a finite dimensional algebra over a field. In this paper it is shown, provided R modulo its radical is commutative, that R has every left indecomposable injective uniserial implies that R is right serial.

\n Dada uma aplicação f: B -> A entre duas C*-álgebras, o cone dessa aplicação, Cf, é o conjunto formado pelos pares (b,g) pertencentes à soma direta da C*-álgebra B com o cone CA tais que f(b) = g(0), sendo CA o cone de A. Neste trabalho estudamos o funtor determinado pela associação da sequência exata curta 0 -> SA -> Cf -> B -> 0 para cada *-homomorfismo f: B -> A, e demonstramos que esse funtor é exato. Caracterizamos as aplicações de conexão associadas à sequência exata 0 -> SA -> Cf -> B -> 0, mostrando que a aplicação do índice é dada por tAK1(f) e que a aplicação exponencial é dada por bAK0(f), sendo tA o isomorfismo entre K1(A) e K0(SA) e bA a aplicação de Bott. Por fim, usando que toda sequência exata curta de C*-álgebras pode ser vista na forma 0 -> Ker f -> B -> A -> 0, mostramos que as aplicações de conexão d1 e d0 associadas a cada sequência exata curta podem ser dadas por dn = Kn+1(j)-1 Kn+1(i) hn, em que j é a inclusão do núcleo de f em Cf, i é a inclusão da suspensão SA também em Cf, hn = bA e h1 = tA .\n

Classically, the Auslander–Bridger transpose finds its best applications in the well-known setting of finitely presented modules over a semiperfect ring. We introduce a class of modules over an arbitrary ring R, which we call Auslander–Bridger modules, with the property that the Auslander–Bridger transpose induces a well-behaved bijection between isomorphism classes of Auslander–Bridger right R-modules and isomorphism classes of Auslander–Bridger left R-modules. Thus we generalize what happens for finitely presented modules over a semiperfect ring. Auslander–Bridger modules are characterized by two invariants (epi-isomorphism class and lower-isomorphism class), which are interchanged by the transpose. Via a suitable duality, we find that kernels of morphisms between injective modules of finite Goldie dimension are also characterized by two invariants (mono-isomorphism class and upper-isomorphism class).

A ring R is called a right WPF-ring (weak PF-ring) if R is a semiperfect right simple-injective ring with essential right socle. The class of right WPF-rings is broader than that of right PF-rings. In this article, we study and provide several characterizations of this new class of rings. We also show that if R is a left perfect, left and right f-injective ring, then R is QF if and only if So c 2(R) is a finitely generated right R-module if and only if R/Soc(R) is a finitely cogenerated left R-module. Some known results are obtained as corollaries.

We show that an iteration of the procedure used to define the Gorenstein projective modules over a ring R yields exactly the Gorenstein projective modules. Specifically, given an exact sequence of Gorenstein projective left R-modules G = … → G 1 → G 0 → G 0 → G 1 → … such that the complex Hom R (G, H) is exact for each projective left R-module H, the module Im(G 0 → G 0) is Gorenstein projective. We also get similar results for Gorenstein flat left R-modules when R is a right coherent ring. As applications, we obtain the corresponding results for Gorenstein complexes.

Let $m \ge 2$ be an integer. By using $m$ submodules of a given module, we construct a certain exact sequence, which is a well known short exact sequence when $m=2$. As an application, we compute a minimal projective resolution of the Jacobson radical of

Let F be a number field, O the ring of algebraic integers in F, and let θ denote the inclusion O ↪ F. The localization exact sequence in algebraic K-theory splits into short exact sequences $$ 0 \to K_n O\xrightarrow{{\theta _\# }}K_n F \to \oplus _m K_{n - 1} \left( {O/m} \right) \to 0 $$ for all positive integers n, where θ # is the homomorphism induced by θ in K-theory and where m runs over the set of maximal ideals of O (see Section 5 of [Q1], Theorem 8 of [Q2] and Theoreme 1 of [S2]); in particular, θ # is always injective. On the other hand, G. Banaszak investigated the subgroup of divisible elements in K n F and explained the important role of these elements in relation with the Lichtenbaum-Quillen conjecture and etale K-theory (see [B1], [B2], [BG], [BZ]). If n is odd, K n F is a finitely generated abelian group and has therefore no non-trivial divisible elements. If n is even, K n F is a large torsion group but all its divisible elements belong to the image of θ # because ⊕ m K n-1 (O/m) is a direct sum of finite cyclic groups and hence contains no non-trivial divisible elements.

Higher homological algebra was introduced by Iyama. It is also known as $n$-homological algebra where $n \geq 2$ is a fixed integer, and it deals with $n$-cluster tilting subcategories of abelian categories. All short exact sequences in such a subcategory are split, but it has nice exact sequences with $n+2$ objects. This was recently formalised by Jasso in the theory of $n$-abelian categories. There is also a derived version of $n$-homological algebra, formalised by Geiss, Keller, and Oppermann in the theory of $( n+2 )$-angulated categories (the reason for the shift from $n$ to $n+2$ is that angulated categories have triangulated categories as the base case). We introduce torsion classes and t-structures into the theory of $n$-abelian and $( n+2 )$-angulated categories, and prove several results to motivate the definitions. Most of the results concern the $n$-abelian and $( n+2 )$-angulated categories ${\mathcal M}( \Lambda )$ and ${\mathcal C}( \Lambda )$ associated to an $n$-representation finite algebra $\Lambda$, as defined by Iyama and Oppermann. We characterise torsion classes in these categories in terms of closure under higher extensions, and give a bijection between torsion classes in ${\mathcal M}( \Lambda )$ and intermediate t-structures in ${\mathcal C}( \Lambda )$ which is a category one can reasonably view as the $n$-derived category of ${\mathcal M}( \Lambda )$. We hint at the link to $n$-homological tilting theory.

A given R-module is called a right np-injective module if for any non-nilpotent element of , and any right R-homomorphism can be extended to , if is np-injective, then is a right np-injective ring. A given ring is called right weakly np-injective if for each non-nilpotent element of , there exists a positive integer such that any right R-homomorphism can be extended to . A given ring is a right weakly np-injective ring, if . In the matrix ring, we poved that is right np-injective, for some , if is right Weakly np-injective. So, We extended many of the known properties and characterizations of right np-injective rings and modules. Finally, the main result in np-injective module found that the R-module is np-injective module if and only if for any the short exact sequence of R-modules, is also a short exact sequence, where and .

In a preceding article~[7] the authors and Tran Ngoc Nam constructed a minimal injective resolution of the mod 2 cohomology of a Thom spectrum. A Segal conjecture type theorem for this spectrum was proved. In this paper one shows that the above mentioned resolutions can be realized topologically. In fact there exists a family of cofibrations inducing short exact sequences in mod 2 cohomology. The resolutions above are obtained by splicing together these short exact sequences. Thus the injective resolutions are realizable in the best possible sense. In fact our construction appears to be in some sense an injective closure of one of Takayasu. It strongly suggests that one can construct geometrically (not only homotopically) certain dual Brown-Gitler spectra.

Throughout this paper, R wilt denote an associative ring with unity, all the R-modu!es RA = A will be left unitat R-modules, and all mappings f : X-+ Y between R-modules will be R-homomorphisms, unless otherwise specified. We shall be concerned with an arbitrary, but fixed class F of R-monomorphisms e: A--~ B. If C.--Coker :~, then we shall say that the arising exact sequence E: 0 . -~A-~ B-~ C--~ 0 is a F-sequence. d will denote the class of the domains of the maps of F, ~ the class of codomains, and ~ the class of the cokernels. An R-module M will be called injective with respect to ~ or E4)~jective if for each homomorphism f : A --=~ M, there exists an f : B-+ M such that f.~, ~ = f We define an R-modute M to be F-injective if M is injective with respect to ~z, ~or each ~sF. This paper wilt examine F-injecti~Jty in general, establishing its properties and pointing out how these properties generalize know.t~ results for previously studied types of injectivity and divisibility~ Examples includfng two new ones are discussed in tI. Sections III and IV are devoted to the question as to when the class of F-injective modules is closed with respect to epimorphic images. Section V gives a necessary and sufficient condition that all R-modules contain maximum F-injective submodules. In VI and VII, respectively~ new concepts of purity and torsionfreeness are defined and studied in relation to F-injectivity,

Exact sequences of fuzzy linear maps