Envelope dimension of modules and the simplified radical formula (Q2862949)

Let \(R\) be a commutative ring and \(M\) a unitary \(R\)-module. The radical (resp. weak radical) of a submodule \(N\) of \(M\) (i.e. the intersection of prime (resp. weakly prime) submodules of \(M\) containing \(N\)) is denoted by \(\mathrm{rad}_{M}(N)\) (resp. \(\mathrm{wrad}_{M}(N)\)). The envelope of a submodule \(N\) of \(M\) is denoted by \(E_{M}(N)\) and defined as the set of all x in M such that \(x=rm\) and \(r^{k}m\) is in \(N\) for some \(m\) in \(M\) and some nonnegative integer \(k\) \textit{R. L. Casland} and \textit{M. E. Moore} [Can. Math. Bull. 29, 37--39 (1986; Zbl 0546.13001)]. The submodule of \(M\) generated by \(E_{M}(N)\) is denoted by \(RE_{M}(N)\) and \(kE_{M}(N)\) denotes the sum of \(k\) copies of \(E_{M}(N)\), where \(k\) is a nonnegative integer. A module \(M\) is said to satisfy (resp. weakly satisfy) the simplified radical formula (s.r.f.) of degree \(k\) if \(\mathrm{rad}_{M}(N)=kE_{M}(N)+N\) (resp. \(\mathrm{wrad}_{M}(N)=kE_{M}(N)+N\)) for each submodule \(N\) of \(M\). This \(k\) gives an indication of how ``close'' \(E(N)\) is to being a submodule (i.e. how ``close'' \(E_{M}(N)\) and \(RE_{M}(N)\) are. The smallest integer \(k\) such that \(RE_{M}(N)=kE_{M}(N)+N\) for each submodule \(N\) of \(M\) is called the envelope dimension of \(M\), denoted by \(\mathrm{edim}M\).NEWLINENEWLINEThe relation between \(\dim M\) (Krull dimension) and \(\mathrm{edim}M\) is investigated. For every ring \(R\), it is shown that \(\dim R \leq \mathrm{edim}R\). For an integral domain \(R\), \(R\) is shown to weakly satisfy the s.r.f (of degree 1) if and only if \(\mathrm{edim}R \leq 1,\) while if \(R\) is a Noetherian ring, then \(R\) (weakly) satisfies the s.r.f. (of degree 1) if and only if \(\mathrm{edim} R \leq 1\) if and only if \(R\) is a ZPI-ring. In the case of a dimension \(k\) chained ring, \(R\) satisfies the s.r.f of degree \(k+1\) and if \(R\) is an integral domain, then \(R\) satisfies the s.r.f. of degree \(k\) (but no lower degree). A formula is found for the \(\mathrm{edim}\) of an artinian ring and such rings are shown to satisfy the s.r.f. of degree equal to the envelope dimension. Noetherians ring such that \(\dim R = \mathrm{edim} < \infty\) are characterized as those that are either a finite direct product of fields or are ZPI-rings of Krull dimension 1. The paper concludes with an example illustrating that there exist rings \(R\) such that \(\mathrm{rad}(N)=RE_{M}(N),\) but which do not satisfy the s.r.f for any nonnegative integer \(k\).