Related papers: Cartan maps and projective modules

Let $R$ be a commutative ring. An $R$-module $M$ is said to be almost projective if ${\rm Ext}^1_R(M, N) = 0$ for any $R_{\mathfrak{m}}$-module $N$ and any maximal ideal $\mathfrak{m}$ of $R$. In this paper, we investigate rings $R$ over…

We address two aspects of finitely generated modules of finite projective dimension over local rings and their connection in between: embeddability and grade of order ideals of minimal generators of syzygies. We provide a solution of the…

We prove an assortment of results on (commutative and unital) NIP rings, especially $\mathbb{F}_p$-algebras. Let $R$ be a NIP ring. Then every prime ideal or radical ideal of $R$ is externally definable, and every localization $S^{-1}R$ is…

Throughout this abstruct $A$ will denote a noetherian commutative ring of dimension $n$. The paper has two parts. Among the interesting results in Part-1 are the following: 1) {\it suppose that $f_1, f_2, ..., f_r$ (with $r \leq n$) is a…

In this paper we study the modules $M$ every simple subfactors of which is a homomorphic image of $M$ and call them co-Kasch modules. These modules are dual to Kasch modules $M$ every simple subfactors of which can be embedded in $M$. We…

Let R be a commutative ring and S be an R-algebra. It is well-known that if N is an injective R-module, then Hom(S,N) is an injective S-module. The converse is not true, not even if R is a commutative noetherian local ring and S is its…

Let $R$ denote a commutative Noetherian (not necessarily local) ring, $M$ an arbitrary $R$-module and $I$ an ideal of $R$ of dimension one. It is shown that the $R$-module $\Ext^i_R(R/I,M)$ is finitely generated (resp. weakly Laskerian) for…

In this paper, we study the problem when a finitely generated torsionless module is projective. Let $\Lambda$ be an Artinian local algebra with radical square zero. Then a finitely generated torsionless $\Lambda$-module $M$ is projective if…

Let $R$ be a commutative Noetherian Henselian local ring. Denote by $\mathrm{mod} R$ the category of finitely generated $R$-modules, and by ${\mathcal G}$ the full subcategory of $\mathrm{mod} R$ consisting of all G-projective $R$-modules.…

Let F be a finitely generated discrete group. Given a covering map H to G of Lie groups with G either compact or complex reductive, there is an induced covering map Hom(F, H) to Hom(F, G). We show that when the fundamental group of G is…

Let $\mathbf{k}$ be a field and let $V: \mathscr{C} \to \mathbf{k}\textup{-Mod}$ be a point-wise finite dimensional persistence modules, where $\mathscr{C}$ is a small category. Assume that for all local Artinian $\mathbf{k}$-algebras $R$…

In this paper we study (non-commutative) rings $R$ over which every finitely generated left module is a direct sum of cyclic modules (called left FGC-rings). The commutative case was a well-known problem studied and solved in 1970s by…

Let $(R,\mm,K)$ be a regular local ring containing a field $k$ such that either char $k=0$ or char $k=p$ and tr-deg $K/\BF_p\geq 1$. Let $g_1,\ldots,g_t$ be regular parameters of $R$ which are linearly independent modulo $\mm^2$. Let…

Let $R$ be a ring (not necessary commutative) with non-zero identity. The unit graph of $R$, denoted by $G(R)$, is a graph with elements of $R$ as its vertices and two distinct vertices $a$ and $b$ are adjacent if and only if $a+b$ is a…

Let $\fa$ be an ideal of a commutative Noetherian ring $R$ with identity and let $M$ and $N$ be two finitely generated $R$-modules. Let $t$ be a positive integer. It is shown that $\Ass_R(H_{\fa}^t(M,N))$ is contained in the union of the…

A commutative ring $R$ is stable if every non-zero ideal $I$ of $R$ is projective over its ring of endomorphisms. Motivated by a paper of Bass in the 1960s, stable rings have received wide attention in the literature ever since then. Much…

Let $R$ be an arbitrary ring with identity and $M$ a right $R$-module with $S=$ End$_R(M)$. In this paper we introduce $\pi$-Rickart modules as a generalization of generalized right principally projective rings as well as that of Rickart…

Let $R$ be a commutative ring with unity. The prime ideal sum graph $\text{PIS}(R)$ of the ring $R$ is the simple undirected graph whose vertex set is the set of all nonzero proper ideals of $R$ and two distinct vertices $I$ and $J$ are…

We introduce a notion of $R$-quadratic maps between modules over a commutative ring $R$ which generalizes several classical notions arising in linear algebra and group theory. On a given module $M$ such maps are represented by $R$-linear…

This paper develops a representation-theoretic approach to the isogeny category $\underline{\mathcal{C}}$ of commutative group schemes of finite type over a field $k$, studied in arXiv:1602:00222. We construct a ring $R$ such that…