MODULES WHOSE PRIMARY-LIKE SPECTRA WITH THE ZARISKI-LIKE TOPOLOGY ARE NOETHERIAN SPACES

Abstract

Let R be a commutative ring with identity and M be a unital R-module.  The primary-like spectrum Spec_L(M) is the collection of all primary-like submodules Q of M such that M/Q is a primeful R-module.  The Zariski-like topology on Spec_L(M), denoted \(\mathcal{T}\), is described by taking the set η = {ν(N) | N is a submodule of M} as the set of closed sets of Spec_L(M), where ν(N) = {Q ∈ Spec_L(M) | \(\sqrt{(N : M)}\) ⊆\(\sqrt{(Q : M)}\).  We establish necessary and sufficient conditions for topological space (Spec_L(M),\(\mathcal{T}\)) to be a Noetherian space.  We show that if M is a finitely generated R-module and |Spec_L(M )| < \(\infty\), then the combinatorial dimension of (Spec_L(M),\(\mathcal{T}\)) and the Krull dimension of R/Ann(M) are equal.  In particular, for the Noetherian space(Spec_L(M),\(\mathcal{T}\)) of zero combinatorial dimension the set of irreducible components is finite, and its elements have the form ν(pM) for some minimal prime ideal p ⊇ Ann(M).