reduced ring

From Wiktionary, the free dictionary
Jump to navigation Jump to search

English

[edit]
English Wikipedia has an article on:
Wikipedia

Noun

[edit]

reduced ring (plural reduced rings)

  1. (algebra, ring theory) A ring R that has no nonzero nilpotent elements; equivalently, such that, for xR, 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 over a ring a reduced module if, for any and , implies . 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).
[edit]

Translations

[edit]