V3I11P2

Prime Spectra and Structural Decomposition of Ideals in Non-Noetherian Commutative Rings

Vishnu Subhash Dhondge^1*

Abstract

The prime spectrum of a commutative ring provides a topological representation of its prime ideals, while ideal decompositions translate algebraic information into geometric structure. In Noetherian rings, ascending-chain conditions guarantee several finiteness properties, including finite irreducible decompositions of closed subsets of the spectrum and finite primary decompositions of ideals. These conclusions need not persist when the Noetherian hypothesis is removed. This paper examines the relationship between minimal prime ideals and irreducible components of closed subsets of Spec (R), with particular attention to the distinction between algebraic decompositions and finite topological decompositions. A criterion is proved: for an ideal I of a commutative ring R, V (I) is a finite union of irreducible closed subsets if and only if I has finitely many prime ideals minimal over it. The proof uses the correspondence between irreducible closed subsets and prime ideals and does not require R to be Noetherian. An explicit example in the polynomial ring k[x1, x2, …] exhibits an ideal with infinitely many minimal primes, showing why finite irreducible decomposition can fail in the non-Noetherian setting. The discussion also separates the existence of a finite prime-spectrum decomposition from the stronger assertion that an ideal admits a finite primary decomposition. The results provide a self-contained structural account of what survives without Noetherianity and identify the precise finiteness point at which the classical theory changes.

Keywords:

Commutative algebra; non-Noetherian rings; prime spectrum; minimal prime ideals; irreducible components; primary decomposition; Zariski topology