Prime Radical and Primary decomposition of Ideals in an L subring

TL;DR

This paper introduces the prime radical concept for ideals in L-rings, proves key properties about radicals and primary ideals, and establishes conditions for primary decomposition in L-rings.

Contribution

It defines prime radical in L-rings, proves its equivalence with other radicals, and provides necessary and sufficient conditions for primary decomposition.

Findings

Prime radicals of an ideal, its radical, and semiprime radical coincide.

For primary ideals, prime radical equals the radical.

Conditions for the existence of primary decomposition in L-rings.

Abstract

In this paper we introduce the concept of a prime radical of an ideal of an L-ring L(mu,R) . Among various results pertaining to this concept, we prove here that prime radicals of an ideal eta, its radical , its semiprime radical S(eta) and its prime radical P(eta) , all coincide. Also we prove that for a primary ideal, its prime radical coincide with its radical. Moreover, we introduce the concept of primary decomposition and reduced primary decomposition of an ideal in an L-ring. We obtain a necessary and sufficient conditions for an ideal of an L-ring to have a primary decomposition. Some more results pertaining to the decomposition of an ideal are established.