Integral and non-integral dimension of modules over non-commutative rings.

In this paper we take a natural homomorphism f M : U Hom R ( M , N ) → HomU ( UM , U N ) where U is an R-Algebra , R=CenU and M , N are any two R-modules.Here we have shown that 1. If P is any projective module then f P is a monomorphism. 2. If G be any generator then f ) ( n G is an epimorphism. 3. If M be a generator and N is any free module then f ) (M TrN is always a monomorphism. 4. If M is a generator and N is any finitely generated free R-module then f ) (M TrN is an isomorphism. 5. If M be any artinian module such that it is subdirect product of simple module then f M is monomorphism iff for each submodule K , f K is monomorphism. 6. Let M be a generator and N is any free module if U is finitely presented then f ) (M TrN is an isomorphism. 7. Let M be any left R-module such that f ) ( Re / N j M M is an isomorphism then f M is an isomorphism.

A ring is quasi-Frobenius if and only if every injective right R-module is projective. Thus, the ring has the property that each right -module embeds in a free module if and only if the ring is a quasi-Frobenius ring. A ring is called right whenever every finitely generated right R-module embeds in a free module. This chapter reviews the FGF which asks whether every right ring is quasi-Frobenius. It is convenient to point out that the problem is a strictly one-sided question, for if the ring is left and right FGF, then Ring is already quasi-Frobenius. If R is a ring such that every cyclic right R-module, and every cyclic left R-module embeds in a free module, then R is quasi-Frobenius. R is right CF whenever every cyclic right R-module embeds in a free module. All rings are associative and with identity, and Mod-R denotes the category of right R-modules.

Preface. Conventions and Notations. 1 An Introduction To Maple. 1.1 The Commands . 1.2 Programming. 2 Linear Systems of Equations and Matrices. 2.1 Linear Systems of Equations. 2.2 Augmented Matrix of a Linear System and Row Operations. 2.3 Some Matrix Arithmetic. 3 Gauss-Jordan Elimination and Reduced Row Echelon Form. 3.1 Gauss-Jordan Elimination and rref. 3.2 Elementary Matrices. 3.3 Sensitivity of Solutions to Error in the Linear System. 4 Applications of Linear Systems and Matrices. 4.1 Applications of Linear Systems to Geometry. 4.2 Applications of Linear Systems to Curve Fitting. 4.3 Applications of Linear Systems to Economics. 4.4 Applications of Matrix Multiplication to Geometry. 4.5 An Application of Matrix Multiplication to Economics. 5 Determinants, Inverses and Cramer s Rule. 5.1 Determinants and Inverses from the Adjoint Formula. 5.2 Determinants by Expanding Along Any Row or Column . 5.3 Determinants Found by Triangularizing Matrices. 5.4 LU Factorization. 5.5 Inverses from rref. 5.6 Cramer s Rule. 6 Basic Linear Algebra Topics. 6.1 Vectors. 6.2 Dot Product. 6.3 Cross Product. 6.4 Vector Projection. 7 A Few Advanced Linear Algebra Topics. 7.1 Rotations in Space. 7.2 Rolling a Circle Along a Curve. 7.3 The TNB Frame. 8 Independence, Basis and Dimension for Subspaces of Rn. 8.1 Subspaces of Rn. 8.2 Independent and Dependent Sets of Vectors in Rn. 8.3 Basis and Dimension for Subspaces of Rn. 8.4 Vector Projection onto a Subspace of Rn. 8.5 The Gram-Schmidt Orthonormalization Process. 9 Linear Maps from Rn to Rm. 9.1 Basics About Linear Maps. 9.2 The Kernel and Image Subspaces of a Linear Map. 9.3 Composites of Two Linear maps and Inverses. 9.4 Change of Bases for the Matrix Representation of a Linear Map. 10 The Geometry of Linear and Affine Maps. 10.1 The Effect of a Linear Map on Area and Arclength in Two Dimensions. 10.2 The Decomposition of Linear Maps into Rotations, Reflections and Rescalings in R2. 10.3 The Effect of Linear Maps on Volume, Area and Arclength in R3. 10.4 Rotations, Reflections and Rescalings in Three Dimensions. 10.5 Affine Maps. 11 Least Squares Fits and Pseudoinverses. 11.1 Pseudoinverse to a Non-Square Matrix and Almost Solving an Overdetermined Linear System. 11.2 Fits and Pseudoinverses. 11.3 Least Squares Fits and Pseudoinverses. 12 Eigenvalues and Eigenvectors. 12.1 What Are Eigenvalues and Eigenvectors, and Why Do We Need Them? 12.2 Summary of Definitions and Methods for Computing Eigenvalues and Eigenvectors as well as the Exponential of a Matrix. 12.3 Applications of the Diagonalizability of Square Matrices. 12.4 Solving a Square First Order Linear. System of Differential Equations ... 12.5 Basic Facts About Eigenvalues and Eigenvectors, and Diagonalizability. 12.6 The Geometry of the Ellipse Using Eigenvalues and Eigenvectors. 12.7 A Maple Eigen-Procedure. Bibliography. Indices. Keyword Index. Index of Maple Commands and Packages.

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.

In 1996, Gersten proved that finitely presented subgroups of a word hyperbolic group of integral cohomological dimension 2 are hyperbolic. We use isoperimetric functions over arbitrary rings to extend this result to any ring. In particular, we study the discrete isoperimetric function and show that its linearity is equivalent to hyperbolicity, which is also equivalent to it being subquadratic. We further use these ideas to obtain conditions for subgroups of higher rank hyperbolic groups to be again higher rank hyperbolic of the same rank. The appendix discusses the equivalence between isoperimetric functions and coning inequalities in the simplicial setting and the general setting, leading to combinatorial definitions of higher rank hyperbolicity in the setting of simplicial complexes and allowing us to give elementary definitions of higher rank hyperbolic groups.

Let K be a field. We obtain conditions for elements of the form {1 + a i, 1 + b j} to generate a free group of rank 2 in a quaternion algebra over K, including the case where the characteristic of the field K is 2, based on an article by Professors Jairo Gonçalves, Arnaldo Mandel and Mazi Shirvani. These results are then applied to obtain pairs of elements that freely generate a free group in many other classes of division rings, such as total fields of fractions of Ore domains and Malcev-Neumann series rings. With these results, we partially answer a conjecture by A. Lichtman, regarding the existence of non-abelian free groups in the multiplicative group of non-commutative division rings.

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.

Certain functional identities on division rings of characteristic two

Due to the theoretical work and computer implementations of, for instance, Barakat, Quadrat and Robertz and their coauthors the theory of finitely generated (f.g.) modules over non-commutative regular noetherian rings of partial differential operators with variable coefficients like the Weyl algebras and over other similar rings has become constructive in recent years. In particular these authors compute the purity or grade filtration of a f.g. module by homological means and discuss its significance for the associated behavior. Pommaret and Quadrat noted this significance already in 1999. In this note it is shown that over an arbitrary commutative noetherian ring of operators the purity filtration of a finitely generated module can be easily computed by means of the the primary decomposition of its zero submodule and indeed, with smaller complexity, by inductively computing equidimensional parts. In most books on Constructive Commutative Algebra the connection between this primary decomposition, the equidimensional parts and the purity filtration is implicitly stated for cyclic modules. For many commutative rings of operators the standard signal modules are injective cogenerators. In this case the purity filtration of the module gives rise to a corresponding filtration of the dual behavior, and the primary decomposition induces additional sum representations of the pure dimensional factors of this filtration. For non-commutative rings of operators the standard signal modules are in general neither injective nor cogenerators, and for such signal modules the usefulness of the purity filtration of the module for the determination of the behavior and its structural properties is not obvious. It is also shown by a counter-example that dimensional purity of the module or behavior does not imply dimensional purity of the initial conditions according to Riquier of the associated homogeneous Cauchy problem.

In this paper all modules are left modules and all module homomorphisms act on the right. Ring homomorphisms are written on the left. If M is a module, let L(M) denote the of submodules of M. The Fundamental Theorem of Projective Geometry asserts that if D and AT are two division rings and X: L(D) -> L(K) is a isomorphism between two three-dimensional free modules, then X is induced by a semilinear isomorphism. This means that there is an additive isomorphism L: D -> KTM and a ring isomorphism a:D K such that (X)L = X(X) for each X e L(DTM) and (dV)L = a(d)(V)L for all V e DTM and deD. For convenience the phrase lattice isomorphism X: A -> 5 will be used to mean X: L(A) -> L(B) is a isomorphism. There has been some interest in generalizing this theorem to larger classes of rings. We prove here:

Direct-Sum Decompositions over Local Rings

Let 𝔻 be an arbitrary division ring and Mn(𝔻) the multiplicative semigroup of all n × n matrices over 𝔻. We describe the general form of non-degenerate homomorphisms from M2(𝔻) to M3(𝔻).

Let D D be a division ring and let V = D n V=D^n be a finite-dimensional D D -vector space, viewed multiplicatively. If G = D ∙ G=D^\bullet is the multiplicative group of D D , then G G acts on V V and hence on any group algebra K [ V ] K[V] . Our goal is to completely describe the semiprime G G -stable ideals of K [ V ] K[V] . As it turns out, this result follows fairly easily from the corresponding results for the field of rational numbers (due to Brookes and Evans) and for infinite locally-finite fields. Part I of this work is concerned with the latter situation, while Part II deals with arbitrary division rings.

Matrix fields over the integers modulo m

Homomorphisms of matrix semigroups over division rings from dimension two to four