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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.
The resulting theorem is sometimes known as the Jacobson–Azumaya theorem. [13] Let J(R) be the Jacobson radical of R. If U is a right module over a ring, R, and I is a right ideal in R, then define U·I to be the set of all (finite) sums of elements of the form u·i, where · is simply the action of R on U. Necessarily, U·I is a submodule of U.
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 ...
Any field is a Jacobson ring. Any principal ideal domain or Dedekind domain with Jacobson radical zero is a Jacobson ring. In principal ideal domains and Dedekind domains, the nonzero prime ideals are already maximal, so the only thing to check is if the zero ideal is an intersection of maximal ideals.
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]
In abstract algebra, Jacobson's conjecture is an open problem in ring theory concerning the intersection of powers of the Jacobson radical of a Noetherian ring.. It has only been proven for special types of Noetherian rings, so far.