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 \(1+\sum_{i=1}^{s}k_i\).
If odd primes p and q are such that p≡3(mod 16) and q≡15(mod 16) or p≡11(mod 16) and q≡7(mod 16) in E_(-pq):y^2=x^3-pqx then, rank 2 can be deduced([2] and [4]). It is trivial that we pursue primes p and q as complex as possible. The primes of the forms p=( ) u^4+( ) v^4+( ) w^4+( ) u^2 v^2+( ) u^2 w^2+( ) v^2 w^2and q=( ) 〖'u〗^4+( ) 〖'v〗^4+( ) 〖'w〗^4+( )^' u^2 v^2+( )^' u^2 w^2+( )^' v^2 w^2 in [5] was the beginning taking more numbers of terms than p=Hu^4+Iu^2 v^2+Kv^4 and q=H〖'u〗^4+I^' u^2 v^2+K^' v^4. In [10], the primes p and q are composed of 22 variables and 253 terms where variables are A and B and C and D and E and F and G and H and I and J and K and L and O and P and Q and R and S and T and U and V and W and Z. In this article, we add more variables to [10].
An element of a unital ring is called strongly not clean if all of its unit multiples fail to be clean and very strongly not clean if all its nonzero products with arbitrary (nonzero) elements fail to be clean. In this paper, we investigate such elements, with a particular focus on matrices over commutative rings. In our main result we show that a $2\times 2$ matrix over any Bézout domain with almost stable range 1 is vsn-clean iff its entries are not coprime, excepting $2I_{2}$.
We appoint that E_pq is an elliptic curve y^2=x^3+pqx with distinct odd primes p and q. If p is supposed as the form p≡1(mod 8) then, there can be given rank at least 2 according to defining prime q. For taking at least 2, generally primes are gotten as p=Hu^4+Iu^2 v^2+Kv^4 and q=H'u^4+I'u^2 v^2+K'v^4⋯⋯(BB) . In this article, we will treat rank of curve E_pq where primes are composed of more terms than (BB).
Set E_(-pqs) as an elliptic curve y^2=x^3-pqsx with different odd primes p and q and s then, we regard rank of this curve.
