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An ideal in a ring is radical if and only if the quotient ring is reduced. The radical of a homogeneous ideal is homogeneous. The radical of an intersection of ideals is equal to the intersection of their radicals: . The radical of a primary ideal is prime. If the radical of an ideal is maximal, then is primary.
The ideal quotient is viewed as a quotient because if and only if . The ideal quotient is useful for calculating primary decompositions. It also arises in the description of the set difference in algebraic geometry (see below). (I : J) is sometimes referred to as a colon ideal because of the notation. In the context of fractional ideals, there ...
This follows from the fact that the quotient module U / V is simple and hence annihilated by J(R). J(R) is the unique right ideal of R maximal with the property that every element is right quasiregular [5] [6] (or equivalently left quasiregular [2]). This characterization of the Jacobson radical is useful both computationally and in aiding ...
The third of the properties listed above says that the set of non-units in a local ring forms a (proper) ideal, [3] necessarily contained in the Jacobson radical. The fourth property can be paraphrased as follows: a ring R is local if and only if there do not exist two coprime proper (left) ideals, where two ideals I 1, I 2 are called coprime ...
A radical class (also called radical property or just radical) is a class σ of rings possibly without multiplicative identities, such that: the homomorphic image of a ring in σ is also in σ. every ring R contains an ideal S (R) in σ that contains every other ideal of R that is in σ. S (R / S (R)) = 0.
Radical of a module. In mathematics, in the theory of modules, the radical of a module is a component in the theory of structure and classification. It is a generalization of the Jacobson radical for rings. In many ways, it is the dual notion to that of the socle soc ( M) of M .
Localization (commutative algebra) In commutative algebra and algebraic geometry, localization is a formal way to introduce the "denominators" to a given ring or module. That is, it introduces a new ring/module out of an existing ring/module R, so that it consists of fractions such that the denominator s belongs to a given subset S of R.
An ideal whose radical is maximal, however, is primary. Every ideal Q with radical P is contained in a smallest P-primary ideal: all elements a such that ax ∈ Q for some x ∉ P. The smallest P-primary ideal containing P n is called the n th symbolic power of P. If P is a maximal prime ideal, then any ideal containing a power of P is P-primary.