When is the idealization R⋉M an 𝒜-ring?

Abstract

We present an answer to a problem raised by Anderson and Chun in [D. D. Anderson and S. Chun, Annihilator conditions on modules over commutative rings, J. Algebra Appl. 16(7) (2017) 1750143] on characterizing when the idealization [Formula: see text] of a ring [Formula: see text] on an [Formula: see text]-module [Formula: see text] is an [Formula: see text]-ring (respectively, an [Formula: see text]-ring) in terms of module-theoretic properties of [Formula: see text] and [Formula: see text]. Also, we are concerned with an open question asked by these two authors which reads the following: What modules over a given ring [Formula: see text] are homomorphic images of modules satisfying the strong Property [Formula: see text]? (see, Question 4.4(1) in the above mentioned paper). This paper highly contributes to answer such a question.