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
Famous quotes containing the words reduced and/or ring:
“The physicians say, they are not materialists; but they are:MSpirit is matter reduced to an extreme thinness: O so thin!—But the definition of spiritual should be, that which is its own evidence. What notions do they attach to love! what to religion! One would not willingly pronounce these words in their hearing, and give them the occasion to profane them.”
—Ralph Waldo Emerson (1803–1882)
“When I received this [coronation] ring I solemnly bound myself in marriage to the realm; and it will be quite sufficient for the memorial of my name and for my glory, if, when I die, an inscription be engraved on a marble tomb, saying, “Here lieth Elizabeth, which reigned a virgin, and died a virgin.””
—Elizabeth I (1533–1603)