A general injectivity for modules

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.

Let R be an associative ring with a unit and N be a left R-module. The set MR(N) = {f: N → N | f(rx) = rf(x), r ∈ R, x ∈ N} is a near-ring with respect to the operations of addition and composition and contains the ring ER(N) of all endomorphisms of the R-module N. The R-module N is endomorphic if MR(N) = ER(N). We call an Abelian group endomorphic if it is an endomorphic module over its endomorphism ring. In this paper, we find endomorphic Abelian groups in the classes of all separable torsion-free groups, torsion groups, almost completely decomposable torsion-free groups, and indecomposable torsion-free groups of rank 2.

Continuing an earlier examination of the codominant dimension of rings and modules, a categorical characterization is given for rings of equivalent dominant and codominant dimensions. Specifically, the question is reduced to when R R and the left minimal injective cogenerator R can be used as test modules respectively for the dominant dimension of the projective modules and the codominant dimension of the injective modules. These conditions are in turn characterized by when the injective projective modules are Z-injective. Also, a new and shortened version is given for the proof that the codominant dimension of the injective modules is equal to the dominant dimension of the projective modules. Throughout, R is an associative ring with unity and all modules considered are unital. We denote a minimal injective cogenerator for R)R by R such being isomorphic to an injective hull of a direct sum of a complete representative class of simple left R-modules. For a left perfect ring R and a left R-module RM we define the codominant dimension of RM, denoted codom dim RM, to be greater than or equal to n if for a minimal projective resolution . P* P M 0, Pi is injective for i = 1, 29, . . ., n. The left codominant dimesnion of R is defined to be codom dim RU. The dominant dimension of RM is defined in a dual manner, and the left dominant dimension of R is dom dimRR. Further, for R left perfect, we define codom dimR9 to be infIcodom dim RE: RE injectivel. Similarly, dom dim Ris given by inf dom dimRP: RP projectivel. Finally, we say that RE (nRP) is a test module for RI (RT) provided codom dim RE= codom dimRI (dom dimR P = dom dimRY). In [2] we showed that dom dim R 9= codom dim RI. As an application it can Presented to the Society, January 25, 1973; received by the editors April 16, 1973. AMS (MOS) subject classifications (1970). Primary 16A60; Secondary 16A50, 16A52.

Let R be an associative ring with unit element. Mod-R and R-Mod will denote the categories of unitary right and left R-modules, respectively, and all modules are assumed to be in Mod-R unless otherwise specified. For all M, N ϵ Mod-R, HomR(M, N) will usually be abbreviated as [M, N]. For the definitions of basic terms, and an exposition on torsion theories in Mod-R, the reader is referred to Lambek [6]. Jans [5] has called a class of modules which is closed under submodules, direct products, homomorphic images, group extensions, and isomorphic images a TTF (torsion-torsionfree) class.

Let $\Lambda$ be a ring endowed with an involution $a \mapsto \tilde{a}$. We say that two units $a$ and $b$ of $\Lambda$ fixed under the involution are congruent if there exists an element $u \in \Lambda^{\times}$ such that $a = ub\tilde{u}$. We denote by $\mathcal{H}(\Lambda)$ the set of congruence classes. In this paper we consider the case where $\Lambda$ is an order with involution in a semisimple algebra $A$ over a local field and study the question of whether the natural map $\mathcal{H}(\Lambda) \rightarrow \mathcal{H}(\Lambda)$ induced by inclusion is injective. We give sufficient conditions on the order $\Lambda$ for this map to be injective and give applications to hermitian forms over group rings.

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.

In this work, we introduce ps-supplement submodules as a kind of supplements. Specifically, a submodule V of a module M is referred to as a ps-supplement if there exists another submodule U⊆M such that M=U+V and the intersection U∩V is projective semisimple. We demonstrate that, over an arbitrary ring, the collection PS of short exact sequences of the form 0⟶M⟶ψN⟶ϕK⟶0, where Im(ψ) is a ps-supplement in N, constitutes a proper class. In addition, we explore several homological aspects of this class, including the symmetry of the relations between the modules. Finally, we provide that a ring R is a left SI-ring with essential socle if and only if every left R-module is ps-supplemented.

Transfer of homological properties under base change is a classical field of study. Let <TEX>$R{\rightarrow}S$</TEX> be a ring homomorphism. The relations of Gorenstein projective (or Gorenstein injective) dimensions of unbounded complexes between <TEX>$U{\otimes}^L_RX$</TEX>(or <TEX>$RHom_R(X,U)$</TEX>) and X are considered, where X is an R-complex and U is an S-complex. In addition, some sufficient conditions are given under which the equalities <TEX>$G-dim_S(U{\otimes}^L_RX)=G-dim_RX+pd_SU$</TEX> and <TEX>$Gid_S(RHom_R(X,U))=G-dim_RX+id_SU$</TEX> hold.

If 0 → A → C → B → 0 is an exact sequence of abelian groups, if ƒ is a 2-cocyle for this extension, if α ∈ End A, and if β ∈ End B, then a necessary and sufficient condition that α extend to an endomorphism γ of C which induces β is that (M) αƒ and ƒβ be cohomologous ; see Montgomery (2). We shall extend this result to the case where 1 → A → G → B → 1 is an exact sequence of groups and A is abelian.

We define and study a notion of Gorenstein projective dimension for complexes of left modules over associative rings. For complexes of finite Gorenstein projective dimension we define and study a Tate cohomology theory. Tate cohomology groups have a natural transformation to classical Ext groups. In the case of module arguments, we show that these maps fit into a long exact sequence, where every third term is a relative cohomology group defined for left modules of finite Gorenstein projective dimension.

This series aims to report new developments in mathematical research and teaching -quickly, informally and at a high level.The type of material considered for publication includes: 1.Preliminary drafts of original papers and monographs 2. Lectures on a new field, or presenting a new angle on a classical field 3. Seminar work-outs 4. Reports of meetings, provided they are a) of exceptional interest or b) devoted to a single topic.

Given an associative ring R with unit we construct the largest homomorphic image of R such that every right invertible square matrix is left invertible.The construction may be accomplished in one step.Related results are also given.In this note all rings are associative with unit element.Such a ring R is said to be weakly finite if, for any square matrices A and B over R such that AB = / (the identity matrix), we also must have BA = I.The terminology is due to P. M. Cohn ([1, 0.2], [2]), but other authors have considered this condition under other names (for example Goodearl,in [3]).If R is not weakly finite, form an ideal / by including the entries of all / -BA where AB = / and A and B are square.Then R/J is closer to being weakly finite, but we may have to repeat the process (possibly infinitely) to obtain finally a weakly finite homomorphic image of R. The purpose of this note is to show that no such repetition is necessary; that is, R/J is itself weakly finite.Other results can be obtained by keeping track of the sizes of the matrices involved.Suppose that A and B are n X n matrices over R and that AB = /.We think of A as a map of free right Ä-modules, and get an exact sequence o^/^/r^/r -0, where P is the kernel of A. The map B provides a splitting of this sequence, and the following result becomes obvious.Proposition 1.Given a right R-module P, the following are equivalent:(i) P © R" s Rn, where Rn denotes the free right R-module of rank n;(ii) there exist n X n matrices A, B over R (considered as maps of free modules)such that AB = I and P ^ \n\(I -BA).Call a projective P stably trivial of rank < n if it satisfies the above condition(s), and call it stably trivial if it is stably trivial of some (finite) rank.Then clearly R is weakly finite if and only if every stably trivial module is the zero module.To keep track of matrix sizes, call R weakly n-finite (terminology of Cohn) if there are no nonzero stably trivial modules of rank < n.Equivalently, R is weakly «-finite if and only if any n X n right invertible matrix over R is left invertible.Recall that the trace ideal t(P) of a projective module P is the ideal generated by the images of Ä-functionals /: P -* R. If P is finitely generated, it is a direct

This article establishes, for an appropriate localisation of associative rings, a long exact sequence in algebraic [math] –theory. The main result goes as follows. Let [math] be an associative ring and let [math] be the localisation with respect to a set [math] of maps between finitely generated projective [math] –modules. Suppose that [math] vanishes for all [math] . View each map in [math] as a complex (of length 1, meaning one non-zero map between two non-zero objects) in the category of perfect complexes [math] . Denote by [math] the thick subcategory generated by these complexes. Then the canonical functor [math] induces (up to direct factors) an equivalence [math] . As a consequence, one obtains a homotopy fibre sequence\n¶\n&lt;math display="block"&gt;\n&lt;mrow&gt;\n&lt;mi&gt;K&lt;/mi&gt;\n&lt;mrow&gt;\n&lt;mo class="MathClass-open"&gt;(&lt;/mo&gt;\n&lt;mrow&gt;\n&lt;mi&gt;A&lt;/mi&gt;\n&lt;mo class="MathClass-punc"&gt;,&lt;/mo&gt;\n&lt;mi&gt;σ&lt;/mi&gt;\n&lt;/mrow&gt;\n&lt;mo class="MathClass-close"&gt;)&lt;/mo&gt;\n&lt;/mrow&gt;\n&lt;mo class="MathClass-rel"&gt;→&lt;/mo&gt;\n&lt;mi&gt;K&lt;/mi&gt;\n&lt;mrow&gt;\n&lt;mo class="MathClass-open"&gt;(&lt;/mo&gt;\n&lt;mrow&gt;\n&lt;mi&gt;A&lt;/mi&gt;\n&lt;/mrow&gt;\n&lt;mo class="MathClass-close"&gt;)&lt;/mo&gt;\n&lt;/mrow&gt;\n&lt;mo class="MathClass-rel"&gt;→&lt;/mo&gt;\n&lt;mi&gt;K&lt;/mi&gt;\n&lt;mrow&gt;\n&lt;mo class="MathClass-open"&gt;(&lt;/mo&gt;\n&lt;mrow&gt;\n&lt;mi&gt;B&lt;/mi&gt;\n&lt;/mrow&gt;\n&lt;mo class="MathClass-close"&gt;)&lt;/mo&gt;\n&lt;/mrow&gt;\n&lt;/mrow&gt;\n&lt;/math&gt;\n¶ (up to surjectivity of [math] ) of Waldhausen [math] –theory spectra.\n¶ In subsequent articles [?, ?] we will present the [math] – and [math] –theoretic consequences of the main theorem in a form more suitable for the applications to surgery. For example if, in addition to the vanishing of [math] , we also assume that every map in [math] is a monomorphism, then there is a description of the homotopy fiber of the map [math] as the Quillen [math] –theory of a suitable exact category of torsion modules.

Let R be an associative ring which is not necessarily commutative. For any torsion theory τ on the category of left R-modules and for any nonnegative integer n we define and study the notion of the nth local cohomology functor with respect to τ. For suitably nice rings a bound for the nonvanishing of these functors is given in terms of the τ-dimension of the modules.

Let R be an associative ring with identity, and let M be a unitary left R-module.The main purpose of this paper is to introduce and study properties of class of modules including lifting module.We use the Radical submodule to introduce this class of modules.We start by defining a generalized lifting module as a generalization of lifting module, then we give another generalization of generalized lifting module we called it cofinitely generalized lifting module.Thus, we give characterization and prove some properties of this type of modules.[