Radical of an ideal

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Definition

Let R be a commutative unital ring and I be an ideal in R. The radical of I, sometimes denoted \sqrt{I}, is defined in the following equivalent ways:

  • It is the set of all a \in R for which some positive power of a lies inside I
  • It is the smallest radical ideal containing I
  • It is the intersection of all prime ideals containing I
  • Under the quotient map R \to R/I, it is the inverse image of the nilradical of R/I

I is a radical ideal iff \sqrt{I} = I.

Equivalence of definitions

After quotienting out, the equivalence of definitions follows from the equivalence of various definitions of the nilradical.