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

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 R ⋉ M of a ring R on an R -module M is an 𝒜 -ring (respectively, an 𝒮 𝒜 -ring) in terms of module-theoretic properties of R and M. Also, we are concerned with an open question asked by these two authors which reads the following: What modules over a given ring R are homomorphic images of modules satisfying the strong Property 𝒜 ? (see, Question 4.4(1) in the above mentioned paper). This paper highly contributes to answer such a question. [ABSTRACT FROM AUTHOR]