Abstract
Abstract Ring properties of amalgamated products are investigated. We offer new, elementary arguments which extend results from [5] and [12] to noncommutative setting and also give new properties of amalgamated rings.
Highlights
Introduction and preliminariesAll rings in this paper are associative, we do not assume they contain unity.We write I A, if I is an ideal of a ring A
: (a) R is closed under extensions, if the following implication holds: I A, I ∈ R and A/I ∈ R =⇒ A ∈ R. (b) R is homomorphically closed, if every homomorphic image of a ring from
R is in R. (c) R is closed under subrings, if every subring of a ring belonging to R is in R. (d) R is hereditary, if I A ∈ R implies that I ∈ R
Summary
Introduction and preliminariesAll rings in this paper are associative, we do not assume they contain unity.We write I A, if I is an ideal of a ring A. : (a) R is closed under extensions, if the following implication holds: I A, I ∈ R and A/I ∈ R =⇒ A ∈ R. The following proposition contains straightforward properties of amalgamated rings. Without loss of generality, in the definition of amalgamated rings we may write f (A) + J instead of B. In the following lemma we present how properties of the class R affect properties of rings A, J and B.
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