An extension of S-noetherian spectrum property

Over a commutative ring the prime spectrum, as a geometrical object, reflects many properties of the ring. This geometrical object can be modified by the action of multiplicative closed subsets, or more in general, by hereditary torsion theories. For a multiplicatively closed subset [Formula: see text] of a commutative ring [Formula: see text], in the literature there were several [Formula: see text]-noetherian spectrum properties. In this paper, for any commutative ring [Formula: see text], we introduce generalizations of them using a hereditary torsion theory [Formula: see text] instead of a multiplicative closed subset [Formula: see text]. Two different constructions of noetherian spectra on a ring, associated to a hereditary torsion theory, are considered in such a way that they are characterized by prime ideals and preserved by polynomial ring constructions.