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

Abstract

Let R be a commutative ring with 1 6= 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 am ∈ I for some a ∈ R implies an ∈ 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. Mathematics Subject Classification (2020): 13F05, 13A15, 13E05, 13F20, 13C10, 13C11, 13F30, 13D05