In ring theory, a ring R 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 A form an ideal of A, the so-called nilradical of A; therefore a commutative ring is reduced if and only if its nilradical is reduced to zero. Moreover, a commutative ring is reduced if and only if the only element contained in all prime ideals is zero.
A quotient ring A/R is reduced if and only if R is a radical ideal.
Read more about Reduced Ring: Examples and Non-examples, Generalizations
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