Let $R$ be a commutative Noetherian ring and $\mathfrak{a}$ a proper ideal of $R$. We show that if $n\,:=\,\text{grad}{{\text{e}}_{R}}\,\mathfrak{a}$, then $\text{En}{{\text{d}}_{R}}(H_{\mathfrak{a}}^{n}(R))\,\cong \,\text{Ext}_{R}^{n}(H_{\mathfrak{a}}^{n}(R),\,R)$. We also prove that, for a nonnegative integer $n$ such that $H_{\mathfrak{a}}^{i}(R)\,=\,0$ for every $i\,\ne \,n$, if $\text{Ext}_{R}^{i}({{R}_{z}},\,R)\,=\,0$ for all $i\,>\,0$ and $z\,\in \,\mathfrak{a}$, then $\text{En}{{\text{d}}_{R}}(H_{\mathfrak{a}}^{n}(R))$ is a homomorphic image of $R$, where ${{R}_{z}}$ is the ring of fractions of $R$ with respect to a multiplicatively closed subset $\{{{z}^{j}}|j\,\ge \text{0}\}$ of $R$. Moreover, if $\text{Ho}{{\text{m}}_{R}}({{R}_{z}},R)=0$ for all $z\,\in \,\mathfrak{a}$, then ${{\mu }_{H_{\mathfrak{a}}^{n}(R)}}$ is an isomorphism, where ${{\mu }_{H_{\mathfrak{a}}^{n}(R)}}$ is the canonical ring homomorphism $R\,\to \,\text{En}{{\text{d}}_{R}}(H_{\mathfrak{a}}^{n}(R))$.

Let $R$ be a dense subring of $\text{End}\left( _{D}V \right)$, where $V$ is a left vector space over a division ring $D$. If $\dim{{\,}_{D}}V\,=\,\infty $, then the range of any nonzero polynomial $f\left( {{X}_{1}},\,\ldots \,,\,{{X}_{m}} \right)$ on $R$ is dense in $\text{End}\left( _{D}V \right)$. As an application, let $R$ be a prime ring without nonzero nil one-sided ideals and $0\,\ne \,a\,\in \,R$. If $af{{\left( {{x}_{1}},\ldots ,{{x}_{m}} \right)}^{n\left( {{x}_{i}} \right)}}\,=\,0$ for all ${{x}_{1}},\,\ldots \,,\,{{x}_{m}}\,\in \,R$, where $n\left( {{x}_{i}} \right)$ is a positive integer depending on ${{x}_{1}},\,\ldots \,,\,{{x}_{m}}\,\in \,R$, then $f\left( {{X}_{1}},\,\ldots \,,\,{{X}_{m}} \right)$ is a polynomial identity of $R$ unless $R$ is a finite matrix ring over a finite field.

Let R be a ring and U a left R-module with S=End(RU). The aim of this paper is to characterize when S is coherent. We first show that a left R-module F is TU-flat if and only if HomR(U,F) is a flat left S-module. This removes the unnecessary hypothesis that U is Σ-quasiprojective from Proposition 2.7 of Gomez Pardo and Hernandez [‘Coherence of endomorphism rings’, Arch. Math. (Basel)48(1) (1987), 40–52]. Then it is shown that S is a right coherent ring if and only if all direct products of TU-flat left R-modules are TU-flat if and only if all direct products of copies of RU are TU-flat. Finally, we prove that every left R-module is TU-flat if and only if S is right coherent with wD(S)≤2 and US is FP-injective.

Let $R$ be a ring and $M_R$ be an $R$-module. We characterize the existence of ${\rm Add} M$-covers and ${\rm Add} M$-envelopes in terms of finiteness conditions of $M$ over its endomorphism ring. We then present some applications related to the existence of well-behaved direct sum decompositions for direct products of copies of $M$. Our results can be viewed as natural extensions of classical theorems of Bass and Chase on coherent and perfect rings.

Let K be a function field over finite field ${\Bbb F}_q$ and let ${\Bbb A}$ be a ring consisting of elements of K regular away from a fixed place ∞ of K. Let φ be a Drinfeld ${\Bbb A}$-module defined over an ${\Bbb A}$-field L. In the case where L is a finite ${\Bbb A}$-field, we study the characteristic polynomial $P$φ(X) of the geometric Frobenius. A formula for the sign of the constant term of $P$φ(X) in terms of ‘leading coefficient’ of φ is given. General formula to determine signs of other coefficients of $P$φ(X) is also derived. In the case where L is a global ${\Bbb A}$-field of generic characteristic, we apply these formulae to compute the Dirichlet density of places where the Frobenius traces have the maximal possible degree permitted by the ‘Riemann hypothesis’.