Let Λ be a left and right noetherian ring and modΛ the category of finitely generated left Λ-modules. In this paper we show the following results: (1) For a positive integer k, the condition that the subcategory of modΛ consisting of i-torsionfree modules coincides with the subcategory of modΛ consisting of i-syzygy modules for any 1 ≤ i ≤ k is left-right symmetric. (2) If Λ is an Auslander rin...

Let R be an hereditary Noetherian prime ring, let S be a "Dedekind closure" of R and let T be the category of finitely generated S-torsion R-modules. It is shown that for all i ~ 0, there is an exact sequence 0 -> Ki(T) -> Ki(R) -> Ki(S) ...... O. If i = 0, or R has finitely many idempotent ideals then this sequence splits. A notion of ''right ideal class group" is then introduced for hereditar...

Given a metric measure space $(X,\mathsf{d},\mathfrak{m})$ and lower semicontinuous, bounded function $k\colon X\to\mathbb{R}$, we prove the equivalence of synthetic approaches to Ricci curvature at $x\in X$ being from below by $k(x)$ in terms $\bullet$ Bakry-\'Emery estimate $\Delta\Gamma(f)/2 - \Gamma(f,\Delta f) \geq k\,\Gamma(f)$ an appropriate weak formulation, curvature-dimension conditio...

Let K m ( 3 ) $$ {K}_m^{(3)} denote the complete 3-uniform hypergraph on vertices and S n {S}_n^{(3)} + 1 n+1 consisting of all 2 \left(\genfrac{}{}{0ex}{}{n}{2}\right) edges incident to a given vertex. Whereas many Ramsey numbers grow either at most polynomially or least exponentially, we show that off-diagonal number r 4 , r\left({K}_4^{(3)},{S}_n^{(3)}\right) exhibits an unusual intermediate...

In this paper, we investigate the Fourier coefficients of eta-quotients forms q λ 0 + l k 1 24 η ( τ ) − , $$ {q}^{\frac{\lambda_0+{l}^k{\lambda}_1}{24}}\eta {\left(\tau \right)}^{-{\lambda}_0}\eta {\left({l}^k\tau \right)}^{-{\lambda}_1}, where \eta \left(\tau \right) is Dedekind eta function, = 5,7 11 13 l=5,7,11,13 and 17; a positive integer, {\lambda}_0,{\lambda}_1 are arbitrary integers. W...

Hyperthyroidism has been associated with atrial fibrillation (AF); however, hyperthyroidism-induced ion channel changes that may predispose to AF have not been fully elucidated. To understand the electrophysiological changes that occur in left and right atria with hyperthyroidism, the patch-clamp technique was used to compare action potential duration (APD) and whole cell currents in myocytes f...

Let R be a ring, a right ideal I of R is called small if for every proper right ideal K of R, I +K = R. A ring R is called right small injective if every homomorphism from a small right ideal to R R can be extended to an R-homomorphism from R R to R R. Properties of small injective rings are explored and several new characterizations are given for QF rings and P F rings, respectively.

Aharoni and Howard and, independently, Huang et al. (2012) proposed the following rainbow version of Erd?s matching conjecture: For positive integers n, k m with n ? km, if each families \(F_{1},\ldots,F_{m}\subseteq\left(\begin{array}{c}[n]\\ k\end{array}\right)\) has size more than \(\max\{\left(\begin{array}{c}n\\ k\end{array}\right)-\left(\begin{array}{c}n-m+1\\ k\end{array}\right),\left(\b...