arXiv subjects

Umamaheswaran Arunachalam

Publications and source records attributed to Umamaheswaran Arunachalam.

4 recordsLinked to original sources

arXiv2026

Pure projective tilting modules associated with a special ring and Goresntein properties

In this paper, we study pure-projective tilting modules and related classes of rings. We introduce the notion of a pure-tilting hereditary ring, namely, a ring over which every ideal is pure-projective tilting, and investigate its structural properties. We prove that a ring R is a pure-tilting hereditary ring if and only if R is hereditary noetherian over a von Neumann regular ring R. In the commutative case, we show that R is a pure one-tilting hereditary ring precisely when R is hereditary noetherian. Using Kaplansky conjecture, we establish a connection between pure-tilting hereditary rings and the hereditary noetherian property of prime factor rings. In category theory, for the torsion pair consisting of Gen of I and the orthogonal class of I in the category of R-modules, we establish that the associated Happel-Reiten-Smalo heart H sub I is a Grothendieck category. We also examine the characterization of Ext-orthogonal classes determined by pure projective tilting modules. In addition, we show that every Gorenstein pure projective tilting module is Gorenstein flat if and only if every Gorenstein pure projective tilting module is strict T-stationary, where T denotes the class of all finitely presented tilting modules. These results establish new links between tilting theory, hereditary ring conditions, and Gorenstein homological structures.

math.RA↗

arXiv2024

Right orthogonal class of pure projective modules over pure hereditary rings

We denote by $\mathcal{W}$ the class of all pure projective modules. Present article we investigate $\mathcal{W}$-injective modules and these modules are defined via the vanishing of cohomology of pure projective modules. First we prove that every module has a $\mathcal{W}$-injective preenvelope and then every module has a $\mathcal{W}$-injective coresolution over an arbitrary ring. Further, we show that the class of all $\mathcal{W}$-injective modules is coresolving (injectively resolving) over a pure-hereditary ring. Moreover, we analyze the dimension of $\mathcal{W}$-injective coresolution over a pure-hereditary ring. It is shown that $\sup\{ \cores_{\mathcal{W}^{\bot}}(M) \colon M \mbox{is an }R\mbox{-module }\} = \Fcor_{\mathcal{W}^{\bot}}(R) = \sup\{\pd(G) \colon G \mbox{ is a pure projective } R\mbox{-module}\}$ and we give some equivalent conditions of $\mathcal{W}$-injective envelope with the unique mapping property. In the last section, we proved the desirable properties of the dimension when the ring is semisimple artinian.

math.RA↗

arXiv2019

Existence of covers and envelopes of a left orthogonal class and its right orthogonal class of modules

In this paper, we investigate the notions of $\mathcal{X}^\bot$-projective, $\mathcal{X}$-injective and $\mathcal{X}$-flat modules and give some characterizations of these modules, where $\mathcal{X}$ is a class of left $R$-modules. We prove that the class of all $\mathcal{X}^\bot$-projective modules is Kaplansky. Further, if the class of all $\mathcal{X}$-projective $R$-modules is closed under direct limits, we show the existence of $\mathcal{X}^\bot$-projective covers and $\mathcal{X}$-injective envelopes over a $\mathcal{X}^\bot$-hereditary ring $R.$ Moreover, we decompose a $\mathcal{X}^\bot$-projective module into a projective and a coreduced $\mathcal{X}^\bot$-projective module over a self $\mathcal{X}$-injective and $\mathcal{X}^\bot$-hereditary ring. Finally, we prove that every module has a $\mathcal{W}$-injective precover over a coherent ring $R,$ where $\mathcal{W}$ is the class of all pure projective modules.

math.AC↗

arXiv2019

Gelfand-Kirillov dimensions of simple modules over twisted group algebras $k \ast A$

For the $n$-dimensional multiparameter quantum torus algebra $Λ_{\mathfrak q}$ over a field $k$ defined by a multiplicatively antisymmetric matrix $\mathfrak q = (q_{ij})$ we show that in the case when the torsion-free rank of the subgroup of $k^\times$ generated by the $q_{ij}$ is large enough there is a characteristic set of values (possibly with gaps) from $0$ to $n$ that can occur as the Gelfand--Kirillov dimension of simple modules. The special case when $\mathrm{K}.\dim(Λ_{\mathfrak q}) = n - 1$ and $Λ_{\mathfrak q}$ is simple studied in A.~Gupta, {\it $\uppercase{\mbox{GK}}$-dimensions of simple modules over $K[X^{\pm 1}, σ]$}, Comm. Algebra, {\bf 41(7)} (2013), 2593--2597 is considered without assuming simplicity and it is shown that a dichotomy still holds for the GK dimension of simple modules.

math.RA↗