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For a general ring with unity R, the Jacobson radical J(R) is defined as the ideal of all elements r ∈ R such that rM = 0 whenever M is a simple R-module.That is, = {=}. This is equivalent to the definition in the commutative case for a commutative ring R because the simple modules over a commutative ring are of the form R / for some maximal ideal of R, and the annihilators of R / in R are ...
If R is commutative, the Jacobson radical always contains the nilradical. If the ring R is a finitely generated Z-algebra, then the nilradical is equal to the Jacobson radical, and more generally: the radical of any ideal I will always be equal to the intersection of all the maximal ideals of R that contain I. This says that R is a Jacobson ring.
Jacobson's conjecture: the intersection of all powers of the Jacobson radical of a left-and-right Noetherian ring is precisely 0. Kaplansky's conjectures; Köthe conjecture: if a ring has no nil ideal other than {}, then it has no nil one-sided ideal other than {}. Monomial conjecture on Noetherian local rings
Any finitely generated algebra over a Jacobson ring is a Jacobson ring. In particular, any finitely generated algebra over a field or the integers, such as the coordinate ring of any affine algebraic set, is a Jacobson ring. A local ring has exactly one maximal ideal, so it is a Jacobson ring exactly when that maximal ideal is the only prime ideal.
It is trivial to show = (using the basic property =), but we give some alternative methods: [clarification needed] The radical corresponds to the nilradical of the quotient ring = [,] / (), which is the intersection of all prime ideals of the quotient ring. This is contained in the Jacobson radical, which is the intersection of all maximal ...
In fact, if M is finitely generated over a ring, then rad(M) itself is a superfluous submodule. This is because any proper submodule of M is contained in a maximal submodule of M when M is finitely generated. A ring for which rad(M) = {0} for every right R-module M is called a right V-ring. For any module M, rad(M/rad(M)) is zero.
If an element of a ring is nilpotent and central, then it is a member of the ring's Jacobson radical. [15] This is because the principal right ideal generated by that element consists of quasiregular (in fact, nilpotent) elements only. If an element, r, of a ring is idempotent, it cannot be a member of the ring's Jacobson radical. [16]
The concept of the Jacobson radical of a ring; that is, the intersection of all right (left) annihilators of simple right (left) modules over a ring, is one example. The fact that the Jacobson radical can be viewed as the intersection of all maximal right (left) ideals in the ring, shows how the internal structure of the ring is reflected by ...