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The factor ring of a radical ideal is a semiprime ring for general rings, and is a reduced ring for commutative rings. Primary ideal: An ideal I is called a primary ideal if for all a and b in R, if ab is in I, then at least one of a and b n is in I for some natural number n. Every prime ideal is primary, but not conversely. A semiprime primary ...
An important ideal of the ring called the Jacobson radical can be defined using maximal right (or maximal left) ideals. If R is a unital commutative ring with an ideal m, then k = R/m is a field if and only if m is a maximal ideal. In that case, R/m is known as the residue field. This fact can fail in non-unital rings.
A subset I is said to be a two-sided ideal or simply ideal if it is both a left ideal and right ideal. A one-sided or two-sided ideal is then an additive subgroup of R . If E is a subset of R , then RE is a left ideal, called the left ideal generated by E ; it is the smallest left ideal containing E .
A ring is a prime ring if and only if the zero ideal is a prime ideal, and moreover a ring is a domain if and only if the zero ideal is a completely prime ideal. Another fact from commutative theory echoed in noncommutative theory is that if A is a nonzero R - module , and P is a maximal element in the poset of annihilator ideals of submodules ...
As an example, the nilradical of a ring, the set of all nilpotent elements, is not necessarily an ideal unless the ring is commutative. Specifically, the set of all nilpotent elements in the ring of all n × n matrices over a division ring never forms an ideal, irrespective of the division ring chosen. There are, however, analogues of the ...
Here is what the most popular diamond shapes for engagement rings mean. While reading, think about which shape reflects you and your partner. Round Brilliant Cut: Traditional and Elegant