(n,Q)-ideals ans phi-(n,Q)-ideals of commutative rings

Technical Summary: (n, Q)-Ideals and $\phi$-(n, Q)-Ideals of Commutative Rings

Problem and Motivation
The paper addresses the need to unify and generalize various classes of ideals in commutative ring theory, specifically those extending the concepts of prime and primary ideals. The author notes that significant developments in this area include the introduction of 2-absorbing ideals by Badawi [3], the subsequent generalization to $n$-absorbing $\delta$-primary ideals by Ulucak et al. [8], and the study of $\phi$-$n$-absorbing primary ideals by Mostafanasab and Darani. Additionally, the literature contains specific classes such as $J$-ideals (related to the Jacobson radical) and $N$-ideals (related to the nilradical), which were later generalized to $\phi$-$(n, I)$ and $\phi$-$(n, N)$ ideals by Khalfi [4] and Anebri et al. [2], respectively. Mimouni [6] previously introduced $Q$-ideals to unify prime, primary, $J$, and $N$ ideals. The central problem this paper tackles is the lack of a single framework that encompasses these diverse generalizations, particularly those involving an arbitrary ideal $Q$ and the function $\phi$.

Methodology
The paper employs a purely algebraic approach within the context of commutative rings with identity. The methodology involves:

1. Definition and Characterization: Introducing new algebraic structures, specifically $(n, Q)$-ideals and $\phi$-$(n, Q)$-ideals, defined by specific conditions on the product of $n+1$ elements.
2. Logical Deduction: Proving theorems that establish relationships between these new ideals and existing classes (e.g., $n$-absorbing, $n$-absorbing primary, $J$-ideals).
3. Structural Analysis: Investigating how these ideals behave under standard ring-theoretic operations, including localization, intersection, and decomposition into direct products of rings.
4. Idealization: Utilizing the concept of idealization ($A(+)M$) to relate properties of ideals in a base ring $A$ to ideals in the idealized ring $A(+)M$.

Key Contributions and Results

• Introduction of $(n, Q)$-ideals: The paper defines an ideal $I$ as an $(n, Q)$-ideal if, for any $a_1 \cdots a_{n+1} \in I$, either $a_1 \cdots a_n \in I$ or there exists an index $t$ such that the product omitting $a_t$ lies in $Q$.

  • It is proven that if $I$ is an $(n, Q)$-ideal, then $I \subseteq Q$.
  • The paper establishes that if $I$ is an $(n, Q)$-ideal for every ideal $Q$ properly containing $I$, then $I$ is an $n$-absorbing primary ideal.
  • A characterization is provided showing that $I$ is an $n$-absorbing primary ideal if and only if it is an $(n, \sqrt{I})$-ideal.
  • Theorem 2.4 demonstrates that an ideal is an $(n, J)$-ideal if and only if it is an $(n, M)$-ideal for every maximal ideal $M$.

• Introduction of $\phi$-$(n, Q)$-ideals: The author generalizes the concept by introducing a function $\phi: I(A) \to I(A) \cup \{\emptyset\}$. An ideal $I$ is a $\phi$-$(n, Q)$-ideal if the condition holds for products in $I \setminus \phi(I)$.

  • The paper proves that if $I$ is a $\phi$-$(n, Q)$-ideal for every $Q$ properly containing $I$, then $I$ is a $\phi$-$n$-absorbing primary ideal.
  • Equivalence conditions are established linking $\phi$-$n$-absorbing primary ideals to $\phi$-$(n, \sqrt{I})$-ideals.

• Structural Properties:

  • Localization: Theorem 2.8 shows that the property of being an $(n, Q)$-ideal is preserved under localization $S^{-1}A$ under specific conditions regarding the multiplicative set $S$ and the zero-divisors of $I$ and $Q$.
  • Intersections: Proposition 2.7 proves that the intersection of $(n_j, Q)$-ideals is an $(n, Q)$-ideal where $n = \sum n_j$.
  • Decomposable Rings: Theorems 2.9 and 2.10 characterize $(n, Q)$-ideals in decomposable rings ($A = A_1 \times \cdots \times A_r$), showing that the property holds globally if and only if it holds component-wise or under specific conditions on the factors.

• Idealization: Theorem 3.5 provides a correspondence between $\phi$-$(n, Q)$-ideals in a ring $A$ and $\psi$-$(n, Q(+)M)$-ideals in the idealization $A(+)M$. This result generalizes Theorem 3.6, which establishes the same correspondence for standard $(n, Q)$-ideals (where $\phi$ and $\psi$ are empty).

Significance
The paper claims to provide a unified framework for several existing generalizations of prime and primary ideals. By introducing $(n, Q)$-ideals and $\phi$-$(n, Q)$-ideals, the author consolidates the study of $n$-absorbing ideals, $n$-absorbing primary ideals, $J$-ideals, $N$-ideals, and their $\phi$-variations into a single theoretical structure. The results demonstrate that many known theorems regarding these specific classes are special cases of the more general theorems presented here. The work extends the theoretical landscape of commutative algebra by offering a common language to discuss these ideal properties and their behavior under ring operations and idealization.