Concept

# Reduced ring

Summary
In ring theory, a branch of mathematics, a ring is called a reduced ring if it has no non-zero nilpotent elements. Equivalently, a ring is reduced if it has no non-zero elements with square zero, that is, x2 = 0 implies x = 0. A commutative algebra over a commutative ring is called a reduced algebra if its underlying ring is reduced. The nilpotent elements of a commutative ring R form an ideal of R, called the nilradical of R; therefore a commutative ring is reduced if and only if its nilradical is zero. Moreover, a commutative ring is reduced if and only if the only element contained in all prime ideals is zero. A quotient ring R/I is reduced if and only if I is a radical ideal. Let \mathcal{N}_R be nilradical of any commutative ring R. There is a natural functor R\mapsto R/\mathcal{N}_R of category of commutative rings \text{Crng} into category of reduced rings \text{Red} and it is left adjoin
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