Amalgamated rings with clean-type properties

Abstract

Let f : A → B be a ring homomorphism and K be an ideal of B. Many variations of the notions of clean and nil-clean rings have been studied by a variety of authors. We investigate strongly π-regular and clean-like properties of the amalgamation ring A ◃▹ f K of A with B along K with respect to f .

Bi-amalgamations subject to j-clean and weakly j-clean properties

“…For example, they proved that if the amalgamation A f J is J-clean, then A and f (A) + J are J-clean. Also they proved that if J ∈ N il(B), and A is J-clean, then A f J is J-clean [16].…”

Amalgamated Rings with n-UJ-Properties

“…In Farshad et al (2021), the authors studied 𝑈𝑈-rings (if 𝑈(𝑅) = 1 + 𝑁𝑖𝑙(𝑅) i.e., rings whose units are unipotent (Calugareanu, 2015) and clean like features of the amalgamation ring 𝐴 ⋈ 𝑓 𝐾 as the followings:…”

“…(3) If 𝐴 ⋈ 𝑓 𝐾 is a nil-clean ring, then 𝐴 and 𝑓(𝐴) + 𝐾 are nil-clean rings (Farshad et al, 2021; Proposition 2.7 (1)) where a ring 𝑅 is clean if each element of 𝑅 is a sum of a unit and an idempotent in 𝑅 (Nicholson, 1977), and 𝑅 is nil-clean if each element of 𝑅 is a sum of a nilpotent and an idempotent in 𝑅 (Diesl, 2013);…”

“…(4) If 𝐴 is nil-clean ring and 𝐾 ⊂ 𝑁𝑖𝑙(𝐵), then 𝐴 ⋈ 𝑓 𝐾 is a nil-clean ring (Farshad et al, 2021;Theorem 2.10 (1)).…”

“…(5) If 𝐴 ⋈ 𝑓 𝐾 is a 𝐽-clean ring, then 𝐴 and 𝑓(𝐴) + 𝐾 are 𝐽-clean rings (Farshad et al, 2021;Proposition 2.16 (1)) where a ring 𝑅 is 𝐽-clean if each element of 𝑅 is a sum of an idempotent and an element from 𝐽(𝑅) (Chen, 2010).…”