ON $(m,n)$-CLOSED IDEALS IN AMALGAMATED ALGEBRA

Abstract

Let R be a commutative ring with 1 = 0 and let m and n be integers with 1 ≤ n < m. A proper ideal I of R is called an (m, n)-closed ideal of R if whenever a m ∈ I for some a ∈ R implies a n ∈ I. Let f : A → B be a ring homomorphism and let J be an ideal of B. This paper investigates the concept of (m, n)-closed ideals in the amalgamation of A with B along J with respect f denoted by A f J. Namely, Section 2 investigates this notion to some extensions of ideals of A to A f J. Section 3 features the main result, which examines when each proper ideal of A f J is an (m, n)-closed ideal. This allows us to give necessary and sufficient conditions for the amalgamation to inherit the radical ideal property with applications on the transfer of von Neumann regular, π-regular and semisimple properties.

On $n$-absorbing prime ideals of commutative rings

“…The concept of n-absorbing ideals is a generalization of the concept of prime ideals (note that a prime ideal of R is a 1-absorbing ideal of R). For more details on n-absorbing ideals, we refer the reader to [11][12][13]. We investigate rings in which every n-absorbing ideal of R is a prime ideal, where n ≥ 2 is an integer, called n-AB rings.…”