Let $(R,\mathfrak m)$ be a commutative Noetherian regular local ring of dimension $d$ and $I$ be a proper ideal of $R$ such that ${\rm mAss}_R(R/I)={\rm Assh}_R(I)$. It is shown that the $R$-module $H^{{\rm ht}(I)}_I(R)$ is $I$-cofinite if and only if
${\rm cd}(I,R)={\rm ht}(I)$. Also we present a sufficient condition under which this condition the $R$-module $H^i_I(R)$ is finitely generated if and only if it vanishes., Jafar A'zami, Naser Pourreza., and Obsahuje bibliografické odkazy
An exchange ring $R$ is strongly separative provided that for all finitely generated projective right $R$-modules $A$ and $B$, $A\oplus A\cong A \oplus B\Rightarrow A\cong B$. We prove that an exchange ring $R$ is strongly separative if and only if for any corner $S$ of $R$, $aS+bS=S$ implies that there exist $u,v\in S$ such that $au=bv$ and $Su+Sv=S$ if and only if for any corner $S$ of $R$, $aS+bS=S$ implies that there exists a right invertible matrix $\begin{pmatrix} a&b\\ *&* \end{pmatrix} \in M_2(S)$. The dual assertions are also proved.