reduced ring

reduced ring (plural reduced rings)

1. (algebra, ring theory) A ring R that has no nonzero nilpotent elements; equivalently, such that, for x ∈ R, x² = 0 implies x = 0.
   • 1997, Thomas G. Lucas, Characterizing When R(X) is Completely Integrally Closed, Daniel Anderson (editor), Factorization in Integral Domains, Marcel Dekker, page 401,

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

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

     Extending the notion of a reduced ring, we call a right module ${\displaystyle M}$ over a ring ${\displaystyle R}$ a reduced module if, for any ${\displaystyle m\in M}$ and ${\displaystyle a\in R}$ , ${\displaystyle ma=0}$ implies ${\displaystyle mR\cap Ma=0}$ . Various results of reduced rings are extended to reduced modules.

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

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

• reduced algebra
• reduced scheme