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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 M is a free module, or if R is right nonsingular, then the converse is true. A semisimple module is nonsingular if and only if it is a projective module. If R is a right self-injective ring, then () = (), where J(R) is the Jacobson radical of R.
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.
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 ...
R is a Jacobson ring; Every prime ideal of R is an intersection of maximal ideals. Every radical ideal is an intersection of maximal ideals. Every Goldman ideal is maximal. Every quotient ring of R by a prime ideal has a zero Jacobson radical. In every quotient ring, the nilradical is equal to the Jacobson radical.
In mathematics, a semi-local ring is a ring for which R/J(R) is a semisimple ring, where J(R) is the Jacobson radical of R. (Lam 2001, p. §20)(Mikhalev & Pilz 2002, p. C.7) The above definition is satisfied if R has a finite number of maximal right ideals (and
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.
This is proven by observing that any nil ideal is contained in the Jacobson radical of the ring, and since the Jacobson radical is a nilpotent ideal (due to the Artinian hypothesis), the result follows. In fact, this can be generalized to right Noetherian rings; this result is known as Levitzky's theorem. [3]