Prime Radical and Primary decomposition of Ideals in an L subring

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.