It is well-known that a based space is of the weak homotopy type of a loop space iff it is a grouplike algebra over an $A_{\infty }$-operad. The classical model for such an operad consists of Stasheff’s associahedra. The present paper describes a similar recognition principle for free loop spaces. Let $𝒫$ be an operad, $M$ a $𝒫$-module and $U$ a $𝒫$-algebra. An $M$-trace over $U$ consists of a space $V$ and a module homomorphism $T:M\to {\text{End}}_{U,V}$ over the operad homomorphism $𝒫\to {\text{End}}_U$ given by the algebra structure on $U$. Let ${𝒞}_1$ be the...

For any Lie-Rinehart algebra $(A,L)$, B(atalin)-V(ilkovisky) algebra structures $\partial$ on the exterior $A$-algebra ${\Lambda }_{A}L$ correspond bijectively to right $(A,L)$-module structures on $A$; likewise, generators for the Gerstenhaber algebra ${\Lambda }_{A}L$ correspond bijectively to right $(A,L)$-connections on $A$. When $L$ is projective as an $A$-module, given a B-V algebra structure $\partial$ on ${\Lambda }_{A}L$, the homology of the B-V algebra $({\Lambda }_{A}L,\partial )$ coincides with the homology of $L$ with coefficients in $A$ with reference to the right $(A,L)$-module structure determined...

Let $(R,𝔪)$ be a complete Noetherian local ring, $I$ an ideal of $R$ and $M$ a nonzero Artinian $R$-module. In this paper it is shown that if $𝔭$ is a prime ideal of $R$ such that $dimR/𝔭=1$ and $\left(0{:}_M𝔭\right)$ is not finitely generated and for each $i\ge 2$ the $R$-module ${\mathrm{Ext}}_R^i(M,R/𝔭)$ is of finite length, then the $R$-module ${\mathrm{Ext}}_R^1(M,R/𝔭)$ is not of finite length. Using this result, it is shown that for all finitely generated $R$-modules $N$ with $Supp\left(N\right)\subseteq V\left(I\right)$ and for all integers $i\ge 0$, the $R$-modules ${\mathrm{Ext}}_R^i(N,M)$ are of finite length, if and only if, for all finitely generated $R$-modules...

Let $R$ be a ring. A right $R$-module $M$ is said to be retractable if $𝕋{Hom}_R(M,N)\ne 0$ whenever $N$ is a non-zero submodule of $M$. The goal of this article is to investigate a ring $R$ for which every right R-module is retractable. Such a ring will be called right mod-retractable. We proved that $\left(1\right)$ The ring ${\prod }_{i\in ℐ}{R}_i$ is right mod-retractable if and only if each $R_i$ is a right mod-retractable ring for each $i\in ℐ$, where $ℐ$ is an arbitrary finite set. $\left(2\right)$ If $R\left[x\right]$ is a mod-retractable ring then $R$ is a mod-retractable ring.