Higher order \(z\)-ideals of special rings (Q2118360)

Let \(R\) be a commutative unitary ring. An ideal \(I\) of \(R\) is called \(z\)-ideal ``if whenever any two elements of \(R\) are contained in the same set of maximal ideals and \(I\) contains one of them, then it also contains the other one.'' If \((A_n)_{n\geq 0}\) is an increasing sequence of commutative rings, \(\mathcal{A}[[X]]\) is defined by \(\mathcal{A}[[X]]=\{\sum_{n\geq 0}a_nX^n\mid a_n\in A_n, \forall n\in\mathbb{N}\}\). A relation between the \(z\)-ideals of \(A_0\) and those of the ring \(\mathcal{A}[[X]]\) is given. If \(R\) is a commutative ring and \(M\) an \(R\)-module let \(R(+) M:=\{\left(\begin{smallmatrix}r &m\\ 0 &r\end{smallmatrix}\right)\mid r\in R,\ m\in M\}\). A description of the \(z\)-ideals of \(R(+) M\) is given too. Moreover there are some results about higher order \(z\)-ideals and it is shown that each SFT-ring is radically \(z\)-covered.