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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]

Examples are given of associative rings in which 1 is the only unit. These rings coincide with the Boolean rings within the universe of one-sided Artinian rings (resp., of von Neumann regular rings; resp., of nonzero algebraic Z2-algebras; resp., of commutative semiquasi-local rings). The class of these rings is stable under direct limits and arbitrary direct products. New examples of such rings include polynomial rings over F2 in an arbitrary (possibly infinite) number of algebraically independent commuting indeterminates. The 0- and 1-generated rings in which 1 is the only unit are classified up to isomorphism. Emphasis is given to the role of the 2-generated one-dimensional integral domains in which 1 is the only unit. Several open questions are identified. [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]