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:
“Realism, whether it be socialist or not, falls short of reality. It shrinks it, attenuates it, falsifies it; it does not take into account our basic truths and our fundamental obsessions: love, death, astonishment. It presents man in a reduced and estranged perspective. Truth is in our dreams, in the imagination.”
—Eugène Ionesco (b. 1912)
“Time has no divisions to mark its passage, there is never a thunderstorm or blare of trumpets to announce the beginning of a new month or year. Even when a new century begins it is only we mortals who ring bells and fire off pistols.”
—Thomas Mann (1875–1955)