A reduced ring is a commutative ring with no nonzero nilpotent element. Equivalently, its nilradical is zero.
If a nonzero nilpotent element existed in , its proper annihilator would lie in a maximal ideal . No element outside annihilates , so is nonzero in the localization at a prime ideal . Conversely, if in a localization, some in the multiplicative subset satisfies . Then , so reducedness of gives and hence .
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In ring theory, a branch of abstract algebra, a **reduced ring** is a type of ring in which there are no non-zero nilpotent elements. A nilpotent element \( a \) in a ring \( R \) is defined as an element such that for some positive integer \( n \), \( a^n = 0 \). In simpler terms, if \( a \) is nilpotent, then raising it to some power eventually results in zero.