On the Interplay Between Property (A) and Ideal Avoidance in Commutative Rings and Modules

Abstract

This paper studies the relation between Property (A) and the ideal avoidance property in commutative rings and modules. Using the annihilator family we reformulate Property (A) as a containment problem for finitely generated ideals inside unions of annihilator ideals. This point of view yields a finite-cover criterion: if a ring has the ideal avoidance property and the zero-divisor set of a module is a finite union of annihilator ideals, then the module has Property (A). We also prove a prime-annihilator criterion, showing that finite covers by prime annihilator ideals force Property (A); as corollaries, every Noetherian module has Property (A), and every reduced ring with finitely many minimal primes has Property (A). A radical version yields nilpotent annihilation from finite covers by prime radicals of annihilator ideals. In addition, we establish invariance under passage to the quotient by  and characterize Property (A) for finite direct products of rings and modules. Examples are included to show that ideal avoidance is sufficient but not necessary, and that the finite-annihilator-cover hypothesis is only a sufficient device. The paper provides a unified framework that links contemporary work on annihilator conditions and ideal avoidance in commutative algebra.