Let 𝓐(ℝ) and 𝓔(ℝ) denote respectively the ring of analytic and real entire functions in one variable. It is shown that if 𝔪 is a maximal ideal of 𝓐(ℝ), then 𝓐(ℝ)/𝔪 is isomorphic either to the reals or a real closed field that is an η₁-set, while if 𝔪 is a maximal ideal of 𝓔(ℝ), then 𝓔(ℝ)/𝔪 is isomorphic to one of the latter two fields or to the field of complex numbers. Moreover, we study the residue class rings of prime ideals of these rings and their Krull dimensions. Use is made of...

On considère l’espace de modules $M(c_1,c_2)$ des fibrés stables de rang $2$ sur ${\mathbb{P}}_k^3$, de classes de Chern $c_1,c_2$, $k$ étant un corps algébriquement clos de caractéristique quelconque. Si ($c_1=0,c_2\>0$) ou ($c_1=-1,c_2=2p\ge 6$), on sait ([7], [9]) que $M(c_1,c_2)$ a une composante irréductible dont le point générique $ℱ(c_1,c_2)$ a la cohomologie naturelle. Nous avons calculé ([16]) la résolution minimale de $ℱ(0,c_2)$. Dans cet article, nous voulons déterminer celle de $ℱ(-1,c_2)$ si ${\displaystyle c_2\>\frac{(v+2)(2v^2+3v-1)}{6v+7},}$ où $v$ est le plus petit entier tel que $h^0\left(ℱ\left(v\right)\right)\>0$. Par un procédé standard rappelé dans [16], on se ramène à des...

We define and study restricted projective, injective and flat dimensions over local homomorphisms. Some known results are generalized. As applications, we show that (almost) Cohen-Macaulay rings can be characterized by restricted homological dimensions over local homomorphisms.

Let $k$ be a field of characteristic zero and $B$ a $k$-domain. Let $R$ be a retract of $B$ being the kernel of a locally nilpotent derivation of $B$. We show that if $B=R\oplus I$ for some principal ideal $I$ (in particular, if $B$ is a UFD), then $B=R^{\left[1\right]}$, i.e., $B$ is a polynomial algebra over $R$ in one variable. It is natural to ask that, if a retract $R$ of a $k$-UFD $B$ is the kernel of two commuting locally nilpotent derivations of $B$, then does it follow that $B\cong R^{\left[2\right]}$? We give a negative answer to this question. The interest in retracts comes...

A ring extension $R\subseteq S$ is said to be FO if it has only finitely many intermediate rings. $R\subseteq S$ is said to be FC if each chain of distinct intermediate rings in this extension is finite. We establish several necessary and sufficient conditions for the ring extension $R\subseteq S$ to be FO or FC together with several other finiteness conditions on the set of intermediate rings. As a corollary we show that each integrally closed ring extension with finite length chains of intermediate rings is necessarily a normal pair...