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reduced ring

noun

Definitions

  1. A ring R that has no nonzero nilpotent elements

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

The neighborhood

Vish — recursive loop

No curated loop yet for reduced ring. Loops are being traced one word at a time while the ingestion pipeline matures.

sense glosses and etymology drawn from English Wiktionary · source · CC-BY-SA