Narrow Results

Let $R$ be a ring. We give the equivalent categories of the module categories over the polynomial ring $R[x]$ and the quotient ring $R[x]/(x^n)$. As applications, we describe explicitly some important classes of modules over $R[x]$ and $R[x]/(x^n)$ such as finitely generated modules, projective modules and injective modules.

An algebraic formulation is given for the embedded noncommutative spaces over the Moyal algebra developed in a geometric framework in [8]. We explicitly construct the projective modules corresponding to the tangent bundles of the embedded noncommutative spaces, and recover from this algebraic formulation the metric, Levi–Civita connection and related curvatures, which were introduced geometrically in [8]. Transformation rules for connections and curvatures under general coordinate changes are given. A bar involution on the Moyal algebra is discovered, and its consequences on the noncommutative differential geometry are described.

We investigate the module theory of a certain class of semilocal rings connected with nearly simple uniserial domains. For instance, we classify finitely presented and pure-projective modules over these rings and calculate their projective dimension.

It is shown that a ring R is right noetherian if and only if every cyclic right R-module is a direct sum of a projective module and a module Q, where Q is either injective or noetherian. This provides an affirmative answer to a question raised by P. F. Smith.

Let R be a ring with identity. We prove that, the flat cover of any simple right R-module is projective if and only if R is semilocal and J(R) is cotorsion if and only if R is semilocal and any indecomposable flat right R-module with unique maximal submodule is projective.

Hereditary rings have been extensively investigated in the literature after Kaplansky introduced them in the earliest 50’s. In this paper, we study the notion of a $\mathrm{\Sigma }$-Rickart module by utilizing the endomorphism ring of a module and using the recent notion of a Rickart module, as a module theoretic analogue of a right hereditary ring. A module $M$ is called $\mathrm{\Sigma }$-Rickart if every direct sum of copies of $M$ is Rickart. It is shown that any direct summand and any direct sum of copies of a $\mathrm{\Sigma }$-Rickart module are $\mathrm{\Sigma }$-Rickart modules. We also provide generalizations in a module theoretic setting of the most common results of hereditary rings: a ring $R$ is right hereditary if and only if every submodule of any projective right $R$-module is projective if and only if every factor module of any injective right $R$-module is injective. Also, we have a characterization of a finitely generated $\mathrm{\Sigma }$-Rickart module in terms of its endomorphism ring. Examples which delineate the concepts and results are provided.

We prove that a ring $R$ has a module $M$ whose domain of projectivity consists of only some injective modules if and only if $R$ is a right noetherian right $V$-ring. Also, we consider modules which are projective relative only to a subclass of max modules. Such modules are called max-poor modules. In a recent paper Holston et al. showed that every ring has a p-poor module (that is a module whose projectivity domain consists precisely of the semisimple modules). So every ring has a max-poor module. The structure of all max-poor abelian groups is completely determined. Examples of rings having a max-poor module which is neither projective nor p-poor are provided. We prove that the class of max-poor $R$-modules is closed under direct summands if and only if $R$ is a right Bass ring. A ring $R$ is said to have no right max-p-middle class if every right $R$-module is either projective or max-poor. It is shown that if a commutative noetherian ring $R$ has no right max-p-middle class, then $R$ is the ring direct sum of a semisimple ring $R_1$ and a ring $R_2$ which is either zero or an artinian ring or a one-dimensional local noetherian integral domain such that the quotient field $Q_2$ of $R_2$ has a proper $R_2$-submodule which is not complete in its $R_2$-topology. Then we show that a commutative noetherian hereditary ring $R$ has no right max-p-middle class if and only if $R$ is a semisimple ring.

A method of constructing (finitely generated and projective) right module structure on a finitely generated projective left module over an algebra is presented. This leads to a construction of a first-order differential calculus on such a module which admits a hom-connection or a divergence. Properties of integrals associated to this divergence are studied, in particular the formula of integration by parts is derived. Specific examples include inner calculi on a non-commutative algebra, the Berezin integral on the supercircle and integrals on Hopf algebras.

An R-module M is said to have the cancellation property provided that M⊕ B ≅ M⊕ C implies B ≅ C for any pair of R-modules B and C. We obtain a characterization of the cancellation property for projective R-modules. With this result, it is proved that Dedekind domains have the cancellation property; and if R is a Prüfer domain, then R⊕ B ≅ R⊕ C implies B ≅ C for any pair of finitely generated R-modules B and C.

The problem of finding all feedback equivalence classes of Brunovsky and locally Brunovsky linear systems defined on a commutative ring is related with combinatorial problem of visiting all partitions of elements in a given monoid.