reduced ring

plural reduced rings

1. ring theoryalgebraA ring R that has no nonzero nilpotent elements; equivalently, such that, for x ∈ R, x² = 0 implies x = 0.

   We do this for reduced rings in Corollary 10, and for rings with nonzero nilpotents in Corollary 15.

   1997, Thomas G. Lucas, “Characterizing When R(X) is Completely Integrally Closed”, in Daniel Anderson, editor, Factorization in Integral Domains, Marcel Dekker, page 401

   2 more examples

   Extending the notion of a reduced ring, we call a right module M over a ring R a reduced module if, for any m#92;inM and a#92;inR, ma#61;0 implies mR#92;capMa#61;0. Various results of reduced rings are extended to reduced modules.

   2004, Tsiu-Kwen Lee, Yiqiang Zhou, “Reduced Modules”, in Alberto Facchini, Evan Houston, Luigi Salce, editors, Rings, Modules, Algebras, and Abelian Groups, Marcel Dekker, page 365

   In general, the first case of importance is the normalization of a reduced ring R in its quotient ring K(R).

   2005, David Eisenbud, The Geometry of Syzygies: A Second Course in Commutative Algebra and Algebraic Geometry, Springer, page 210