An associated sequence of ideals of an increasing sequence of rings: z-ideals, prime spectrum and Krull dimension

Abstract

Let [Formula: see text] be an increasing sequence of unitary commutative rings, [Formula: see text], and [Formula: see text] an indeterminate over [Formula: see text]. The sequence [Formula: see text] is said to be an associated sequence of ideals of [Formula: see text], if [Formula: see text] and for each [Formula: see text], [Formula: see text] is an ideal of [Formula: see text] contained in [Formula: see text]. We denote [Formula: see text] In this paper, we deal with the ring [Formula: see text]. We start the paper by studying the [Formula: see text]-ideals of this ring. In fact, we give a full description of the [Formula: see text]-ideals of the ring [Formula: see text]. Next, we study the prime spectrum of [Formula: see text], and finally, we close the paper by some results about it Krull dimension.