Envelopes of commutative rings (Q1758010)

For a class of objects \(\mathcal F\) in a given category \(\mathcal C\) and an object \(X\in\mathcal C\), a morphism \(f:X\longrightarrow F\in\mathcal F\) is called an \(\mathcal F\)-pre-envelope of \(X\), if every other morphism \(f:X\longrightarrow F'\in\mathcal F\) factors through \(f\). If, in addition, \(f\) is (left) minimal (i.e. every endomorphism \(v:F\longrightarrow F\) with \(vf=f\) must be an isomorphism), then \(f\) is an \(\mathcal F\)-envelope of \(X\). The notions of \(\mathcal F\)(pre)cover are defined dually. These notions were introduced by \textit{E. E. Enochs} [Isr. J. Math. 39, 189--209 (1981; Zbl 0464.16019)] as generalizations of injective envelope and projective covers respectively. Given the category CRings of commutative rings with unities, its subclass \(\mathcal F\) and a commutative ring \(R\), the authors' aim is to find precise conditions that would ensure \(R\) has an \(\mathcal F\)-envelope. Some of the results obtained are as follows: (1) If \(\mathcal F\) consists of fields, then \(R\) has an \(\mathcal F\)-envelopes iff \(R\) is a local ring of Krull dimension 0 and the envelopes are of the form \(R\longrightarrow R/\mathbf m\) (\(\mathbf m\) maximal). (2) If \(\mathcal F\) consists of semisimple rings, then rings \(R\) have \(\mathcal F\)-envelopes iff Spec(\(R\)) is finite and the envelopes are of the form \(R\longrightarrow \prod_{{\mathbf p}\in \text{Spec}(R)}k(\mathbf p)\). (3) If \(\mathcal F\) consists of integral domains, then rings \(R\) have \(\mathcal F\)-envelopes iff Nil(\(R\)) is a prime ideal and the envelopes are of the form \(R\longrightarrow R_{\mathrm{red}}=R/\text{Nil}(R)\). More complicated cases are treated as well: One result states that if \(R\) is a ring with a preenvelope that is Noetherian, then \(R\) satisfies the ACC on radical ideals and \(\mathrm{Spec}(R)\) is a Noetherian topological space with the Zariski topology. A ring of Krull dimension zero has a Noetherian (pre)envelope iff it is a finite direct product of local rings which are Artinian modulo the infinite radical. A ring \(R\) has an epimorphic Noetherian envelope iff it has a nil ideal \(I\) such that \(R/I\) is Noetherian and \(\mathbf pI_{\mathbf p}=I_{\mathbf p}\), for all \({\mathbf p}\in \text{Spec}(R\)). The problem of determining the commutative rings which have a Noetherian envelope is shown to be reducible to the problem of identifying those which have a monomorphic Noetherian envelope The authors conjecture that there is no non-Noetherian commutative ring with a monomorphic Noetherian envelope.