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”, in 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”, in 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