Classical Zariski topology of modules and spectral spaces. II. (English) Zbl 1195.16003

This paper is a sequel to the authors’ previous study on a Zariski topology for the spectrum of prime submodules of a module over an associative ring [part I, Int. Electron. J. Algebra 4, 104-130 (2008; Zbl 1195.16002)]. The patch topology is considered, and using a characterization by M. Hochster [Trans. Am. Math. Soc. 142, 43-60 (1969; Zbl 0184.29401)], sufficient conditions are given for the spectrum of a module to be spectral – that is, homeomorphic to the prime spectrum of a commutative ring. In particular, Noetherian modules and Artinian modules over PI rings are spectral. It is further proved that the maximal spectrum of a Noetherian module is homeomorphic to the maximal spectrum of a commutative ring.


16D10 General module theory in associative algebras
16D25 Ideals in associative algebras
16W80 Topological and ordered rings and modules
54H13 Topological fields, rings, etc. (topological aspects)
16P40 Noetherian rings and modules (associative rings and algebras)
16S38 Rings arising from noncommutative algebraic geometry
16D80 Other classes of modules and ideals in associative algebras