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Let A be an Archimedean f-ring with identity and assume that A is equipped with another multiplication ∗ so that A is an f-ring with identity u. Obviously, if ∗ coincides with the original multiplication of A then u is idempotent in A (i.e., u 2 = u ). Conrad proved that the converse also holds, meaning that, it suffices to have u 2 = u to conclude that ∗ equals the original multiplication on A. The main purpose of this paper is to extend this result as follows. Let A be a (not necessary unital) Archimedean f-ring and B be an ℓ -subgroup of the underlaying ℓ -group of A. We will prove that if B is an f-ring with identity u, then the equality u 2 = u is a necessary and sufficient condition for B to be an f-subring of A. As a key step in the proof of this generalization, we will show that the set of all f-subrings of A with the same identity has a smallest element and a greatest element with respect to the inclusion ordering. Also, we shall apply our main result to obtain a well known characterization of f-ring homomorphisms in terms of idempotent elements. [ABSTRACT FROM AUTHOR]

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 $n-1$. [ABSTRACT FROM AUTHOR]

Let R be a domain with quotient field K and prime subring A. Then R is integral over each of its subrings having quotient field K if and only if (A;R) is a survival pair. This shows the redundancy of a condition involving going-down pairs in a earlier characterization of such rings. In characteristic 0, the domains being characterized are the rings R that are isomorphic to subrings of the ring of all algebraic integers. In positive (prime) characteristic, the domains R being characterized are of two kinds: either R = K is an algebraic field extension of A or precisely one valuation domain of K does not contain R. [ABSTRACT FROM AUTHOR]

In this paper, we construct derived equivalences between subalgebras of some -Auslander-Yoneda algebras from a class of triangles in idempotent complete triangulated categories. The derived equivalences are obtained by transferring subalgebras induced by triangles to endomorphism algebras induced by approximation sequences. [ABSTRACT FROM AUTHOR]

Let ν be a (discrete) valuation of the rational function field k(t), where k is a field, and let [Cscr]ν be the intersection of the valuation rings of all the valuations of k(t), other than that of ν. It is well-known that either ν = νπ, the valuation determined by some irreducible polynomial π in k[t], or ν = ν∞, the “valuation at infinity”. In this paper we prove that GL2([Cscr]ν), where ν = νπ, is the fundamental group of a certain tree of groups. The tree has finitely many vertices and its terminal vertices correspond with the elements of the ideal class group of [Cscr]ν. This extends a previous result of Nagao for the special case ν = ν∞. In this case [Cscr]ν = k[t] and Nagao proves that GL2(k[t]) is an amalgamated product of a pair of groups. As a consequence we show that, when the degree of π is at least 4, GL2([Cscr]ν) has a free, non-cyclic quotient whose kernel contains (for example) all the unipotent matrices. This represents a two-dimensional anomaly. [ABSTRACT FROM AUTHOR]