Modules with Chain Conditions on Non-essential Submodules

Abstract

We investigate when modules which satisfy the descending (respectively, ascending) chain condition on non-essential submodules are uniform or Artinian (respectively, Noetherian).     

Polynomial modules and a generalized cauchy problem for systems of holomorphic partial differential operators

In this paper, we study holomorphic solutions of overdetermined systems of partial differential equatoins with constant coefficients. We parametrize the solution space, using an ideal of holomorphic functions generated by certain polynomials associated to the principal symbols of the operators, by proving a Cauchy-Kowalevsky type theorem for these systems. In this theorem, we introduce a non-characteristicity condition for systems which is a direct generalization of a condition for single equations introduced by the author and H.S. Shapiro in a previous paper.

As a tool for studying the systems mentioned above, certain estimates, which may be of independent interest, for elements of analytic modules generated by polynomials are obtained.     

On the transposition anti-involution in real Clifford algebras I: the transposition map

A particular orthogonal map on a finite-dimensional real quadratic vector space (V,?Q) with a non-degenerate quadratic form Q of any signature (p,?q) is considered. It can be viewed as a correlation of the vector space that leads to a dual Clifford algebra C?(V*,?Q) of linear functionals (multiforms) acting on the universal Clifford algebra C?(V,?Q). The map results in a unique involutive automorphism and a unique involutive anti-automorphism of C?(V,?Q). The anti-involution reduces to reversion (resp. conjugation) for any Euclidean (resp. anti-Euclidean) signature. When applied to a general element of the algebra, it results in transposition of the element matrix in the left regular representation of C?(V,?Q). We also give an example for real spinor spaces. The general setting for spinor representations will be treated in part II of this work [R. Ab?amowicz and B. Fauser, On the transposition anti-involution in real Clifford algebras II: Stabilizer groups of primitive idempotents, Linear Multilinear Algebra, to appear].     

Complements to Projective Almost Complete Tilting Modules

ABSTRACT

An equivalent version of the Generalized Nakayama Conjecture states that any projective almost complete tilting module admits a finite number of non-isomorphic indecomposable complements. Motivated by this connection, we investigate the number of possible complements of projective almost complete tilting modules for some particular classes of Artin algebras, namely monomial algebras and algebras with exactly two simple modules.     

Plastic Plate Bending Problem with Friction on the Boundary and Uncertain Input Data

A thin plate made of a plastic material obeying the Hencky–Mises stress-strain law is considered. On a part of the boundary displacements and/or rotations with friction are prescribed. The model leads to a variational inequality of the second kind with a nonlinear monotone operator. Since the material function, loading, and friction coefficients are uncertain, we employ the worst scenario method. The existence of a worst scenario and a convergence of approximate solutions is proved.     

Lifting Property of Direct Sums of Hollow Modules

We give necessary and sufficient conditions for a module M with a decomposition that complements maximal direct summands to be a lifting module. We apply our results to get a necessary and sufficient condition for a direct sum of modules to be a quasi-discrete (discrete) module. We also give some sufficient conditions for a lifting module to be a direct sum of indecomposables.     

Generalization of Nilpotency of Ring Elements to Module Elements

We define nilpotent and strongly nilpotent elements of a module M and show that the set 𝒩 _s (M) of all strongly nilpotent elements of M over a commutative unital ring R coincides with the classical prime radical β _cl (M) the intersection of all classical prime submodules of M.     

𝔏-Rickart Modules

It is well known that the Rickart property of rings is not a left-right symmetric property. We extend the notion of the left Rickart property of rings to a general module theoretic setting and define 𝔏-Rickart modules. We study this notion for a right R-module M _R where R is any ring and obtain its basic properties. While it is known that the endomorphism ring of a Rickart module is a right Rickart ring, we show that the endomorphism ring of an 𝔏-Rickart module is not a left Rickart ring in general. If M _R is a finitely generated 𝔏-Rickart module, we prove that End _R (M) is a left Rickart ring. We prove that an 𝔏-Rickart module with no set of infinitely many nonzero orthogonal idempotents in its endomorphism ring is a Baer module. 𝔏-Rickart modules are shown to satisfy a certain kind of nonsingularity which we term “endo-nonsingularity.” Among other results, we prove that M is endo-nonsingular and End _R (M) is a left extending ring iff M is a Baer module and End _R (M) is left cononsingular.