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We present a proof that all straight domains are locally divided—thereby answering two open problems posed by Dobbs and Picavet, which appeared in the survey "Open Problems in Commutative Ring Theory" written by Cahen, Fontana, Frisch, and Glaz. In fact, we are able to prove a stronger result: a prime ideal of a domain is straight if and only if it is locally divided. [ABSTRACT FROM AUTHOR]

Let R be a commutative ring with nonzero identity. A proper ideal I of R is called a 1-absorbing prime ideal (respectively, 1-absorbing primary ideal) if whenever nonunit elements a , b , c ∈ R with a b c ∈ I , then a b ∈ I or c ∈ I (respectively, a b ∈ I or c ∈ I ). The purpose of this paper is to study the transfer of certain 1-absorbing-like properties to amalgamation of A with B along J with respect to f (denoted by A ⋈ f J), introduced and studied by D'Anna, Finocchiaro and Fontana. Our results provide new techniques for the construction of new original examples satisfying the above-mentioned properties. [ABSTRACT FROM AUTHOR]