Weak Ideal Invariance and Localisation

Rings where the annihilators ofα-critical modules are prime ideals

“…In addition, for a ring R with Krull dimension this property is true under any one of the following conditions; (i) R is weakly ideal invariant (ii) R satisfies the left AR-condition, (iii) the prime ideals of R are right localizable. For right Noetherian ring, the conditions (i) and (iii) are shown to imply this property in [5], For Noetherian AR-rings the same is true from [5] and [13]. We extend the results of K. Brown, T. H. Lenagan, and J. T. Stafford [5] for (i), (ii), and (iii) to rings with Krull dimension.…”

“…For right Noetherian ring, the conditions (i) and (iii) are shown to imply this property in [5], For Noetherian AR-rings the same is true from [5] and [13]. We extend the results of K. Brown, T. H. Lenagan, and J. T. Stafford [5] for (i), (ii), and (iii) to rings with Krull dimension. The proofs are short and direct, utilizing the procedures of [2] and [4].…”

“…If R is a Noetherian ring with both the right and left ARcondition, the result follows from 3.4 of [5], which is a consequence of 3.4 of [13]. PROPOSITION …”

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“…We remark that it is an open question whether or not A is a Prime ring if A is supposed to be a Noetherian ring and, moreover, that has a finitely generated, faithful, critical right module M. This question is equivalent to the question of whether or not the nilradical of A is weakly ideal invariant. For this question and related results see [2,7]. We mention that the above question is open even when we assume that A has an Artinian quotient ring.…”