All rings considered in this paper are supposed to be integral domains. The ÿeld of fractions of a ring R is denoted k(R); the Krull dimension dim R; and the integral closure R : For a ring extension R ⊂ S; we denote by [R; S] the set of all rings T such that R ⊆ T ⊆ S, tr.deg[S : R] the transcendence degree of S over R, and by R * the integral closure of R in S. If the set [R; S] is ÿnite, we will denote by |[R; S]| its cardinality. If P and Q are two primes of R such that P ⊆ Q; then [P; Q] denote the set of all primes˝of R such that P ⊆˝⊆ Q:The main purpose of this paper is to study residually algebraic extensions and pairs. The ring extension R ⊂ S is said to be residually algebraic, if for each prime Q of S and P = Q ∩ R; the extension R=P ⊂ S=Q is algebraic. The pair of rings (R; S) is said to be residually algebraic if for each ring T in [R; S]; the extension R ⊂ T is residually algebraic. This kind of extensions were ÿrst considered by D.E. Dobbs and M. Fontana to characterize homomorphisms of universally incomparable rings [6]. They were also recently considered by A. Ayache and P.J. Cahen [3] for the study of rings R such that the extension R ⊂ T satisÿes the dimension inequality for each overring T of R; i.e. T ∈ [R; k(R)]:In the ÿrst and second sections, we settle the elementary properties of residually algebraic extensions and pairs. We also compare them with extensions that are going up, incomparable, lying over, integral, going down, satisfying the dimension formula or the dimension inequality, Propositions 1.5 and 1.6, Corollary 1.7, Theorem 2.3, Corollary 2.11, and Theorem 2.12. We give also several characterizations of residually algebraic pairs. We show that such pairs satisfy the (u; u −1 )-lemma, which enable us to give a surprising generalization of this lemma, Theorem 2.5. A global version of this characterization is also settled, Theorem 2.10. In the third section we impose some ÿniteness conditions on the ring R, such as to be semilocal and of ÿnite

Let $X$ be a nonempty set and ${\mathcal{P}}(X)$ the power set of $X$. The aim of this paper is to identify the unital subrings of ${\mathcal{P}}(X)$ and to compute its cardinality when it is finite. It is proved that any topology $\unicode[STIX]{x1D70F}$ on $X$ such that $\unicode[STIX]{x1D70F}=\unicode[STIX]{x1D70F}^{c}$, where $\unicode[STIX]{x1D70F}^{c}=\{U^{c}\mid U\in \unicode[STIX]{x1D70F}\}$, is a unital subring of ${\mathcal{P}}(X)$. It is also shown that $X$ is finite if and only if any unital subring of ${\mathcal{P}}(X)$ is a topology $\unicode[STIX]{x1D70F}$ on $X$ such that $\unicode[STIX]{x1D70F}=\unicode[STIX]{x1D70F}^{c}$ if and only if the set of unital subrings of ${\mathcal{P}}(X)$ is finite. As a consequence, if $X$ is finite with cardinality $n\geq 2$, then the number of unital subrings of ${\mathcal{P}}(X)$ is equal to the $n$th Bell number and the supremum of the lengths of chains of unital subalgebras of ${\mathcal{P}}(X)$ is equal to $n-1$.

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