Weakly 2-nil primary ideals of commutative rings

We study the effect of restricting the \(2\)-nil primary condition to nonzero triple products and introduce the resulting class of weakly \(2\)-nil primary ideals. The distinction between the weak and non-weak notions is characterized by \(2\)-nil primary zero triples. We also examine the behavior of these ideals under homomorphisms, quotient rings, localization, and direct products. In integral domains, an ideal \(I\subsetneq R\) satisfies the weakly \(2\)-nil primary condition exactly when \(\sqrt I\) is prime. Hence, in this case, such ideals are quasi-primary, and the weak and non-weak \(2\)-nil primary conditions agree. This also yields a complete description in principal ideal domains. We further determine the weakly \(2\)-nil primary ideals of \(\mathbb Z_n\). If \(n=p_1^{k_1}\cdots p_s^{k_s}\), then for \(s\leq2\), all proper ideals of \(\mathbb Z_n\) satisfy the weakly \(2\)-nil primary condition. For \(s\geq3\), the weakly \(2\)-nil primary ideals are exactly \[ (0)\quad\text{and}\quad(p_i^t), \qquad 1\leq i\leq s,\quad 1\leq t\leq k_i. \] Hence, in this case, their number is \(1+\sum_{i=1}^{s}k_i\).